The existence of antimatter was first proved in 1932 by Carl Anderson. Anderson utilized the property of magnetic fields that moving positive and negative particles bend in opposite different directions. He showed that there was a particle with the mass of an electron that moved in the opposite direction from the electrons in an magnetic field, meaning that the particle was of positive charge, the antimatter equivalent of electrons, the positron. The discovery of the antiproton took longer than the discovery of the positron because mass of a proton is two thousand times greater than that of an electron. Therefore, much more energy is necessary to make an antiproton bend like a positron. The existence of antimatter can be proven using electromagnets in the above described manner, or by using accelerators to produce antimatter and letting the antimatter and matter collided together, annihilate, and release their rest mass energy energy. It is now believed that every fundamental particle has a corresponding antiparticle.
Understanding antimatter requires understanding three main ideas. The first idea is the theory of relativity, with E = mc2. Relativity tells us that no particle can travel faster than light, and that everything is time relative - event viewed from one frame of reference and from another frame moving with respect to it do not observe to physically separated events to occur in the same relative time. The second important idea lies in quantum mechanics. Both the uncertainty principles and wave properties must be understood to completely grasp antimatter. And the last idea is causality, meaning the cause comes before and effect. While these three ideas may not seem integral to antimatter, knowing them is essential to understanding antimatter and how it works. Antimatter is a complete symmetric mirror of regular matter. They both have the same chemical and physical properties. For example, antimatter ice would have the exact same melting temperature as regular ice. One difference, though, is that antiparticles are equivalent to ordinary matter particles moving backwards through time. This is where the three ideas come in. While a particle should not be able to affect things that light which started at the same place and time cannot affect, there is always a chance that it could, because of the uncertainty principle. However, when and if this ever happens, some interactions with an antiparticle will push it back into the correct space.
Science fiction often makes use of antimatter, such as a method of energy storage. The truth is, matter and antimatter collisions are perfectly efficient, meaning all the mass would be converted into pure energy. In this case, antimatter would be the ideal way of transporting energy. When an particle and an antiparticle meet (for example, an electron and a positron), they annihilate, leaving two photons. The matter and antimatter would be annihilated in equal amounts in this case, and to give some idea of how powerful this is, 60 kilograms of mass converted into pure energy would be the equivalent of thousands of thermonuclear warheads.
As temperature increases for matter, the matter can be seen as becoming less complicated, but more symmetric. For example, ice has complicated crystal structures, but not much symmetry as compared to water, which is at a higher temperature. In turn, water vapor is simpler and more symmetric than water in liquid phase. This idea continues as the temperature goes up, eventually with the matter breaking from individual molecules into individual atoms, then nuclei break into free electrons and nuclei, protons and electrons, mesons, and finally quarks and antiquarks. At this extremely high temperature (which is representative of the temperature of the big bang), everything would be simple and have perfect symmetry. Every quark in this matter would have a symmetric antiquark to balance it. It is believed that at the beginning of the universe, all matter was at this highest state, made up of equal numbers of quarks and antiquarks. The next question is, how did the matter and antimatter separate or get out of perfect balance and collect into regions dominated by matter as the universe expanded and cooled down?
Upon simple observation of the universe it becomes apparent that the majority of the observable universe (if not all of it) is made of regular matter. Although there have been many searches, there appears to be no substantial amount of antimatter in the universe. So what would happen to the symmetric amounts of matter and antimatter created during the big bang? It becomes apparent now that there must have been a slight asymmetry in the amounts of matter and antimatter. This asymmetry may have been as small as one part in a billion. So, in the early universe, there may have been what some physicists call "a matter-antimatter war", a billion particles would annihilate with a billion anti-particles leaving one particle.
So, the search for anti-matter continues. Experiments have been launched in both high-altitude balloons and the Space Shuttle to try to determine whether cosmic rays bombarding the Earth originate from matter or antimatter. So far the results have been negative. One interesting source of anti-matter occurs in vacuum energy. While classical physics tends to believe that interstellar space is a true vacuum, with no matter present. Yet, quantum mechanics allows for the possibility that "virtual" particle-antipartice pairs can be created out of the vacuum. This particle-antiparticle pair can exist for a period of time limited by the Heisenberg uncertainty principle. Shortly after creation of this pair, they rejoin and annihilate each other. This still does not provide an explanation for existence for large amounts of antimatter.
An interesting article
A link to a page describing the HEAT (High-Energy Antimatter Telescope) experiment
A Fermilab site about Cosmic Antimatter Detection
Balloon-borne antimatter (anti-carbon and anti-oxygen primarily) search
Astromag on Space Station and its successor AMS (Alpha Magnetic Spectrometer)
Is there a cloud of antimatter at the center of the Galaxy?