Antimatter (sometimes called "Contra-terrene" or "Seetee") is weird stuff that explodes if it comes into contact with ordinary matter. Specifically if atom of antimatter comes into contact with an atom of matter, the mass of both is converted from matter into energy. And by "explode" we mean "makes a nuclear bomb look like a damp firecracker." This process is technically called "annihilation" which is such a vividly evocative term.
Any Star Trek fan can tell you it can't be beat when it comes to getting the most bang for your buck. How much bang? Well, in theory if you mix one gram of matter with one gram of antimatter you should get 1.8×1014 joules of energy or about 43 kilotons.
Why 1.8×1014 joules? Surely you remember Einstein's famous E = Mc2. c is the speed of light which is 299,792,458 meters per second. Squared it is 89,875,517,900,000,000 or about 9.0×1016. M is mass in kilograms and E is energy in joules. So 0.002 kilograms (2 grams) times 9.0×1016 equals 1.8×1014 joules. QED.
Once more, to get some idea of the amount of damage represented by a given amount of Joules, refer to the Boom Table.
Technically, it is the matter and antimatter subatomic particles that annhilate each other, not the atoms as such. Antimatter protons are called anti-protons, antimatter electrons are called positrons, and antimatter neutrons are called anti-neutrons.
So if an anti-atom of anti-hydrogen (with one positron and one anti-proton) strikes a normal atom of helium-4 (with two electrons, two protons, and two neutrons) the single positron will annihilate one electron and the single anti-proton will annihilate one proton. The remaining electron, proton, and two neutrons will come flying out of the blast. Actually the energy of the explosion will create other particles as well, see below.
The antimatter version of a given particle looks like the particle seen in a mirror (that is, some of the properties have equal magnitude but opposite sign). The practical effect is the particle has the opposite charge, a proton has a positive charge while an anti-proton has a negative charge. Neutrons have no charge, neither do anti-neutrons. But anti-neutrons spin in the opposite direction and have other differences.
If you read scientific papers about antimatter, they sometimes denote antiparticles by writing a bar over the particle's symobol. So if a proton is p an antiproton is written as p. Just to be annoying, sometimes they denote antiparticles by writing the charge sign. Electrons have a negative charge, positrons have a postive charge. So they are written as e- and e+ respectively. Older scientific papers are even more annoying. Some use the term "negatron" to mean "electron", other use it to mean "anti-proton". Thankfully the term is now pretty much obsolete.
The main virtue of antimatter power is that it is incredibly concentrated, which drastically reduces the mass of antimatter fuel required for a given application. And mass is always a problem in spacecraft design, so any way of reducing it is welcome. Every gram counts.
What is not well known is that unless the situation is non-standard, antimatter is not a power source. It is an energy transport mechanism. Let me explain.
The same situation with antimatter also exists with respect to the so-called "hydrogen economy". Proponents point out how hydrogen is a "green" fuel, unlike nasty petroleum or gasoline. Burn gasoline and in addition to energy you also produce toxic air pollution. Burn hydrogen and the only additional product is natural ecologically pure water.
The problem is that while there exist petroleum wells, there ain't no such thing as a hydrogen well. You can't find hydrogen just lying around somewhere, the stuff is far too reactive. Hydrogen has to be generated by some other process. The generation process consumes energy (such as electrolysing water using electricity generated by a coal-fired power plant). This is why hydrogen is not a fuel, it is an energy transport mechanism. It is basically being used to transport the energy from the coal-fired power plant into the hydrogen burning automobile. Sort of like miles of copper electrical wires converted into a cryogentic tank of fuel.
This means that unless there exist "antimatter mines", antimatter is also an energy transport mechanism, not a power source.
In Star Trek, I believe they found drifts of antimatter in deep space which made convenient antimatter mines. In real life, astronomers haven't seen many matter-antimatter explosions (signature of antimatter mines). Well, they've seen a few 511 keV gamma rays (the signature of electron-positron antimatter annihilation), but they've all been from thousands of light years away and most seem to be associated with large black holes. If they are antimatter mines, they are most inconveniently located. In Jack Williamson's novels Seetee Ship and Seetee Shock there exist commercially useful chunks of antimatter in the asteroid belt. However, if this was actually true, I think astronomers would have noticed all the antimatter explosions detonating in the belt by now.
The man known as magic9mushroom drew my attention to the fact that Dr. James Bickford has identified a sort of antimatter mine where antimatter can be collected by magnetic scoops (be sure to read the comment section), but the amounts are exceedingly small. He foresees using tiny amounts of antimatter for applications such as catalyzing sub-critical nuclear reactions, instead of just using raw antimatter for fuel. His report is here.
Dr. Bickford noted that high-energy galactic cosmic rays (GCR) create antimatter via "pair production" when they impact the upper atmospheres of planets or the interstellar medium. Planets with strong magnetic fields enhance antimatter production. One would think that Jupiter would be the best at producing antimatter, but alas its field is so strong that it prevents GCR from impacting the Jovian atmosphere at all. As it turns out, the planet with the most intense antimatter belt is Earth, while the planet with the most total antimatter in their belt is Saturn (mostly due to the rings). Saturn receives almost 250 micrograms of antimatter a year from the ring system. Please note that this is a renewable resource.
Dr. Bickford calculates that the plasma magnet scoop can collect antimatter about five orders of magnitude more cost effective than generating the stuff with particle accelerators.
Keep in mind that the quantities are very small. Around Earth the described system will collect about 25 nanograms per day, and can store up to 110 nanograms. That has about the same energy content as half a fluid ounce of gasoline, which ain't much. However, such tiny amounts of antimatter can catalyze tremendous amounts of energy from sub-critical fissionable fuel, which would give you the power of nuclear fission without requiring an entire wastefully massive nuclear reactor. Alternatively, one can harness the power of nuclear fusion with Antimatter-Catalyzed Micro-Fission/Fusion or Antimatter-Initiated Microfusion. Dr. Bickford describes a mission where an unmanned probe orbits Earth long enough to gather enough antimatter to travel to Saturn. There it can gather a larger amount of antimatter, and embark on a probe mission to the outer planets.
So, no antimatter mines means antimatter is an energy transport mechanism. The next problem is that antimatter is a very inefficient energy transport mechanism. Current particle accelerators have an abysmal 0.000002% efficiency in converting electricity into antimatter (I don't care what you saw in the movie Angels and Demons). The late Dr. Robert Forward says this is because nuclear physicist are not engineers, an engineer might manage to increase the efficiency to something approaching 0.01% (one one-hundredth of one percent). Which is still pretty lousy. It means for every megawatt of electricity you pump in to the antimatter-maker you would only obtain enough antimatter to create a mere 100 pathetic watts. The other 999,900 watts are wasted.
The theoretical maximum efficiency of converting electricity into antimatter is 50% due to the pesky Law of Baryon Number Conservation (which demands that when turning energy into matter, equal amounts of matter and antimatter must be created).
In Charles Pellegrino and George Zebrowski novel The Killing Star they deal with this by having the Earth government plate the entire equatorial surface of the planet Mercury with solar power arrays, generating enough energy to produce a few kilograms of antimatter a year (and enough waste heat to make the entire planet start to vaporize). They do this with von Neumann machines, of course. The novel needed antimatter fuel, because when you are trying to delta V a starship up to 96% c and back down, every microgram counts.
On the subatomic particle level, the antimatter version of a proton is an antiproton. An antimatter electron is called a positron, and an antimatter neutron is an anti-neutron.
You might have the mistaken idea that when you mix antimatter and matter that you get energy. That turns out not to be the case.
First off, a particle will only annihilate with the corresponding anti-particle. This means if an electron hits an anti-proton, they will just bounce off each other (actually, protons and antineutrons sometime annihilate, and vice versa. But that does not happen very often).
Electron-positron annihilations do turn into energy, in the form of gamma rays. But note that electrons and positrons are approximately 1/1836 the mass of protons and other nucleons. So if you are mixing atoms of anti-hydrogen with atoms of hydrogen, the electrons and positrons will contribute about 1/1836th of the resulting energy. Electrons and positrons have a mass of 9.10938291×10-31 kilograms, so an electron-positron annihilation produces about 1.6×10-15 joules.
The trouble is with proton-antiproton annihilations. This produces (on average) two neutral and three charged pions. And energy that manifests itself in the fact that the particles are moving at very high velocities.
The neutral pions almost instantly decay into gamma rays. Which is good if you want gamma rays. If you don't they are an inconvenient blast of deadly radiation traveling in all directions.
The charged pions (traveling at 0.94c) will move about 21 meters from the reaction before decaying into muons and neutrinos. The fact they are charged means they can be directed by electrostatic fields for propulsion or their energy harvested by electromagnetic fields to generate electricity.
The charged particles are annoying if you are trying to make an antimatter bomb. 21 meters from ground zero the charged particles will decay into muons and neutrinos that will do no damage whatsoever to the target. This means about 30% of the energy of the antimatter bomb is wasted.
Unsurprisingly, it is very difficult to safely contain antimatter. When antimatter touches matter you get an earth-shattering kaboom. Unfortunately conventional fuel tanks are made of matter. And if you make the fuel tanks out of antimatter the problem becomes: how do you attach them to the matter structural members of your spacecraft?
The strategy is to use electromagnetic or electrostatic energy fields instead of matter walls to hold the blasted stuff.
Earnshaw's theorem proves that no set of static charges can be used to create a stable trap. The best you can do is metastable, and the vast majority of configurations are actively unstable. You need to cheat with nonstationary dynamic fields, as in a Penning Trap.
The report Alternate Pathways to Antimatter Containment by J.M. Rejcek et. al. is relevant to our interests
First off, the report suggests the factors that can be used to rate various types of antimatter containment:
A given system might not have all of these, but the factors are useful for ranking systems in order of desireablity.
Penning traps are the current containment system of choice. It uses electrostatic fields to confine clouds of positrons or anti-protons (both of which are charged).
The trouble is that like charges repel, so the more anti-protons you try to cram into a Penning trap, the more the cloud wants to expand due to electrostatic repulsion, and the more energy you'll need for the confining electrostatic fields to keep it from rupturing. Which is alarming, since the words "rupturing" and "antimatter" are so often seen with words like "blast radius" and "no survivors."
At some point the energy you'll need for the confining field will be more than the energy you'll get from the antimatter, which sort of defeats the purpose of it being a power source. A simplistic estimate is this comes at about 4.4×1012 positrons, which would yield about 0.7 joules.
You can avoid the electrostatic repulsion problem by using uncharged antimatter (i.e., anti-hydrogen atoms), but then you cannot use electrostatic fields to contain it. Clouds of anti-hydrogen atoms cannot be contained by magnetic fields, and we don't know how to make artificial gravity fields. Cant' use matter containers either, so the question of what to use is rather burning.
Magnetic Levitation can be used with solid or liquid antimatter, as long as it is diamagnetic. This will work with solid or liquid anti-hydrogen and anti-lithium. Problems include convincing positrons and anti-protons to combine into liquid anti-hydrogen, making the magnetic field tight enough so no anti-atoms escape, making the system stable, making the system safe, and figuring out how the heck to pour or dump liquid/solid antimatter from one container to another. The magnetic fields in the two containers will tend to interfere with each other, creating a magnetic "pipe" between the two containers is incredibly difficult, and moving the antimatter through the pipe without losing a single anti-atom is non-trivial.
Dynamic Fields are time varying external electric and magnetic fields that in theory can hold antimatter particles. Positronium is an "exotic atom" consisting of an electron and a positron chasing each others tail. Ordinarily it has an average lifespan of 125 picoseconds (trillionths of a second), but a 1997 report said crossed magnetic and electric fields could be used to stabilize it. Other scientists say the report is full of it, still others say that the positronium will still have a "drift velocity" which will let the positronium self destruct anyway.
Stabilized Molecular Bound States are a type of positron-molecule bound states. Under certain circumstances, a molecule of matter can hold a positron for a few whole nanoseconds (billionths of a second) before it blows up. So some scientist speculate that if they wish real hard it might not be impossible that there exists a bound state that will hold a positron for years. Maybe. Hopefully.
Matter Storage means using ordinary solid-state matter as sort of a "sponge" to store antimatter. In science fiction this appears in Schlock Mercenary by trapping antihelium atoms inside fullerene molecules.
Such a sponge would have to have  stable potential wells that will bind positive or negative charges and  wells deep and wide spaced enough to minimize the quantum overlap between the antimatter particles and the matter particles.
Proposed materials include silicate minerals such as zeolite clays, compounds with micro scale pores, nanotubes, and fullerenes. As far as energy density goes, if C60 fullerenes had 1 in ten containing a captured positron, the energy density would be about 2.7×1020 positrons per cubic centimeter, about 6,000 times that of TNT. You would think that this violates Earnshaw's theorem, but quantum mechanics rears its ugly head and fangs the heck ouf of Earnshaw. Actually, if you ignore quantum mechanics, Earnshaw's theorem predicts that hydrogen atoms are impossible, which obviously is not true.
I did stumble over a patent for trapping anti-protons in fullerenes.
In Michael McCollum's novel Thunder Strike! antimatter is transported in torus-shaped Penning traps, they are used to alter the orbits of asteroids ("torus" is a fancy word for "donut").
Dr. Robert Forward spoke of storing antimatter in the form of a frozen snowball of anti-hydrogen at temperatures below two Kelvin, levitated in a magnetic field to avoid contact with the chamber wall. In a vacuum, of course. The cold temperature is to keep the blasted stuff from sublimating any anti-atoms from the surface and starting an annihilation reaction with the chamber. There will be some infrequent annihilation events caused by stray cosmic rays, but these should not be a problem.
If you are using your ball of antimatter as a fuel source instead of a bomb, Dr. Forward suggests extracting antimatter fuel from the chamber by using ultraviolet lasers. The lasers ionize a bit of anti-hydrogen from the snowball, which is captured by tailored electrostatic fields and piped to the engine. To insure the snowball's mass is not removed asymmetrically (which would destabilize the magnetic levitation), it is spun on its axis while under the laser.
Converting the energy from antimatter annihilation into electricity is also not very easy.
The electrons and positrons mutually annihilate into gamma rays. However, since an electron has 1/1836 the mass of a proton, and since matter usually contains about 2.5 protons or other nucleons for each electron, the energy contribution from electron-positron annihilation is negligible. You could use pure positrons, if you are willing to put up with the fact that you'll need 1836 times as many of the little suckers as compared to anti-protons, for the same energy released. You'll need more fullerenes.
Attempting to efficiently convert gamma rays into electricity is left as an exercise for the reader.
For every five proton-antiproton annihilations, two neutral pions are produced and three charged pions are produced (that is, 40% neutral pions and 60% charged pions). The neutral pions almost immediately decay into gamma rays. The charged pions (with about 94% the speed of light) will travel 21 meters before decaying into muons. The muons will then travel an additional two kilometers before decaying into electrons and positrons.
This means your power converter needs a component that will transform gamma rays into electricity, and a second component that has to attempt to extract the kinetic energy out of the charged pions and convert that into electricity. The bottom line is that there is no way you are going to get 100% of the annihilation energy converted into electricity. Exactly what percentage is likely achievable is a question above my pay grade.
Alternatively, as previously mentioned, tiny amounts of antimatter can catalyze tremendous amounts of energy from sub-critical fissionable fuel. This which would give you the power of nuclear fission without requiring an entire wastefully massive nuclear reactor. In the same manner, one can harness the power of nuclear fusion with Antimatter-Catalyzed Micro-Fission/Fusion or Antimatter-Initiated Microfusion
There are quite a few schemes that attempt to harness antimatter for spacecraft propulsion.
The two antimatter weapons I've run across are explosive antimatter warheads and particle beam weapons using antimatter.
An antimatter particle beam will do some impressive damage to the target. But if the particles are moving faster than about 90% c, you will have about the same energy release if the partcles are matter or antimatter. At relativistic velocities antimatter particles are a waste of money and effort.
Antimatter warheads have many problems.