Matter is the stuff you are made of.
Antimatter is the same as matter in every way, looks the same, behaves the same, except its particles have electrical charges opposite to matter. E.g., our electrons are negatively charged, whereas a positron (an antimatter "electron") is positively charged. The positron is the "anti-particle" of the electron.
When a particle meets its anti-particle, they "annihilate": the two particles disappear, and gamma photons are released carrying off their energy. For this reason, should a lump of matter touch a lump of antimatter, they would annihilate, and a giant explosion would result because of the huge energy released (E=mc^2).
Matter and antimatter are definitely related: same thing, but with opposite signs. Twins, but opposites.
It is not clear why, but it seems like there isn't that much antimatter out there, more like trace amounts. Definitely not as much as regular matter as far as we can tell. This is puzzling to physicists and cosmologists, because you'd expect the Big Bang to make roughly equal amounts of matter and antimatter. Scientists agree that the paradox of "excess matter" will advance physics even further once it's solved.
Dark matter - we don't really know what it is. It's not even sure it's "matter" in a conventional sense, or related to it in any way. We just know that galaxies are rotating in such a way that indicates there's a lot more mass out there, but it is mass that we cannot see and cannot be accounted for in the usual ways. Hence the name "dark" (as in invisible) matter.
Dark matter doesn't seem to interact much with regular matter, except gravitationally. Right now dark matter could be passing through you and you wouldn't notice. Dark matter also does not interact with light, so you can't see it. It doesn't seem to interact much with itself either, so for this reason dark matter cannot form "clumps" such as planets or stars. Instead, it probably exists in a diffuse form. Bottom line, dark matter interacts pretty much only via gravity.
The shape of galaxies is a proof of the existence of dark matter, and is a result of the interaction between matter and dark matter. Without dark matter, galaxies would be much less massive, and the outer parts would rotate much more slowly compared to the center. Due to dark matter, galaxies are quite massive, and they rotate almost as solid objects - the outer parts rotate approximately as fast as the central parts.
Estimates vary, but it seems like there's something like 5x to 6x more dark matter out there compared to regular matter.