Every elementary particle in the Universe appears to have a partner particle
called its antiparticle that shares many of the same characteristics, but many
other characteristics are the opposite of those for the particle. For example,
the electron has as its antiparticle the antielectron. The electron and the
antielectron have exactly the same masses, but they have exactly opposite
The common stuff around us appears to be "matter", but we routinely produce antimatter in small quantities in high energy accelerator experiments. When a matter particle meets its antimatter particle they destroy each other completely (the technical term is "annihilation"), releasing the equivalent of their rest masses in the form of pure energy (according to the Einstein E=mc^2 relation). For example, when an electron meets an antielectron, the two annihilate and produce a burst of light having the energy corresponding to the masses of the two particles.
Because the properties of matter and antimatter parallel each other, we believe that the physics and chemistry of a galaxy made entirely from antimatter would closely parallel that of our our matter galaxy. Thus, is is conceivable that life built on antimatter could have evolved at other places in the Universe, just as life based on matter has evolved here. (But if your antimatter twin should show up some day, I would advise against shaking hands---remember that matter and antimatter annihilate each other!) However, we have no evidence thus far for large concentrations of antimatter anywhere in the Universe. Everything that we see so far seems to be matter. If true, this is something of a mystery, because naively there are reasons from fundamental physics to believe that the Universe should have produced about as much matter as antimatter.
As one simple example of the evidence for dark matter, the velocity of rotation for spiral galaxies depends on the amount of mass contained in them. The outer parts of our own spiral galaxy, the Milky Way, are rotating much too fast to be consistent with the amount of matter that we can detect; in fact the data indicates that there must be about 10 times as much matter as we can see distributed in some diffuse halo of our galaxy to account for its rotation. The same is true for most other spiral galaxies where the velocities can be measured.
There are various candidates for the dark matter, ranging from ordinary matter that we just can't see because it isn't bright enough (for example, ordinary matter bound up in black holes, or very faint stars, or large planet-like objects like Jupiter) to more exotic particles that have yet to be discovered. There are some fairly strong arguments based on the production of the light elements in the Big Bang indicating that the majority of the dark matter cannot be ordinary matter or antimatter (which physicists call "baryonic matter"), and thus that the majority of the mass of the Universe is in a form very different from the matter that makes up us and the world around us (physicists call this "non-baryonic matter"). If that is true, then the matter that we are made of (baryonic matter) is but a small impurity compared to the dominant matter in the universe (non-baryonic matter). As someone has put it, "not only are we not the center of the Universe, we aren't even made of the right stuff!"
The nature of the dark matter is perhaps the most fundamental unsolved problem in modern astronomy.