To provide some background, Dirac first discovered when formulating his equation of relativistic quantum mechanics (although not a field theory) that just as many negative energy states existed as eigenfucntions as did positive energy states. Left to his own interpretation he decided that these negative energy states were a sort of sea of negative energy states that were already filled by nature (somehow). However we now interpret these states as anti-particles, that is those particles with the same quantum numbers, except opposite charge, as of "normal" fermions. On some level this distinction is arbitrary as the universe is symmetric wrt matter/antimatter modulo CP violating effects. Carl Anderson then discovered the anti-electron or positron in 1932 in a bubble chamber experiment. This was the first instance of humanity "knowing" antimatter existed.
Nowadays anti-matter is commonly seen in cosmic rays events, being produced in particle accelerator collisions, and even is produced for use in particle accelerators. For example at the Fermi National Accelerator Lab (www.fnal.gov) the particle accelerator named the Tevatron uses anti-protons produced by colliding 150 GeV protons into a nickel target, which are then collected if in the correct energy range and stored in another ring called the accumulator. These anti-protons (or pbars as most physicists call them) are then collided in the Tevatron with protons at a center of mass energy of 1.96 TeV. In these collisions themselves physicists at particle detector experiments like CDF (www-cdf.fnal.gov) observe the production of an entire array of anti-matter particles from positrons to anti-neutrinos to anti-B mesons.
As far as anti-matter being abundant in "other" places in the universe, physics is currently struggling with this. While in experiment we have always observed particles and anti-particles produced in pairs, we observe a much higher fraction of particles than anti-particles in the universe via cosmic ray detection etc. We also do not measure significant amounts of matter anti-matter annhilation via gamma rays spectrum measurements from experiments like Fermi Gamma Ray Space Telescope and ground based experiments like Auger etc. But such a significant difference in the amounts of matter and anti-matter seems to contradict our observation of particles being only produced in pairs (except in special cases such as single top prodcution etc. however here the quantum numbers are still conserved which is the actual important point). This apparent violation is thought to be able to be explained by strong CP violation however this has not been observed in nature as of yet.
If you want to learn more about particle physics you should visit Fermilab in Batavia, ill. It is the nation's leading particle physics lab which is open to the public from dawn to dusk everyday. (www.fnal.gov). Also it has a really neat herd of bison, but that is completely tangential.