As swansont has already stated, dark matter is matter that is hypothesized to exist due to the apparent gravitational effects we see, but which doesn't interact electromagnetically like the regular matter we are used to dealing with. This means it does not emit, reflect, or absorb light either, thus the description "dark".
And as he said, we don't really know just what it is made up of yet.
There are a couple of possibilities. Massive Compact Halo Objects (MACHOs) would be objects like neutron stars and black holes. These are object formed from "normal" matter (or at least a far as black holes go initially from it), but compact so much mass into a small area that they are just to small and dim to see individually. However, we need quite a bit of dark matter to explain observations, and there are reasons why we don't think the universe could have this much mass in the form of MACHOs, as it would have effected how the universe evolved, resulting in one that looks a bit different from what we see.
Another possibility is Weakly Interacting Massive Particles (WIMPs) These are sub-atomic particles that have a rest mass, but just don't interact via the electromagnetic interaction. (And pretty much everything having to do with how we interact with regular matter, from touching it or seeing it, to chemical reactions involves electromagnetic interaction)
These "ghost-like" particles would pass right through you like you weren't even there. While this this seems bizarre, we actually already know of a particle that behaves like this, the neutrino; Billions of them pass through you every day with your ever noticing it. Neutrinos ( or at least the type we know of) don't work for dark matter for various reasons, so the WIMPs of dark matter would be something like, but not identical to neutrinos.
It is also possible that both of the above play a role in making up the total effect we see.
swansont also mentioned attempts to explain things by a modified theory of gravity. The problem with this is that a number of observations are not compatible with such an explanation. An example would be galaxies that appear to be identical but exhibit different gravity profiles. Even if the rule of gravity were different than what we presently think they are, they would still need to be consistent from galaxy to galaxy. So while its perfectly possible for different galaxies to contain different amounts of dark matter and thus as a whole act differently in terms of gravity, it is hard to explain why the actual rules governing gravity would change between galaxies.
Having said this, it is still possible for a modified gravity theory to play some role, if combined with dark matter. A new theory of gravity which also incorporates a mix of MACHOs and WIMPs could end up being the final answer.
Right now we are at the stage of continuing to make observations in order to narrow the playing field. We have eliminated some possibilities, but there are more to explore.