It should fire in space with certain reservations...
In the pressurized environment of the bullet casing/gun, it would work just fine. Gunpowder has its own oxidizer built in, so it needs no air to function. However, a loosely packed gunpowder charge (or any other similar low explosive) would not really burn very well. This has nothing to do with its ability to burn, and much more to do with the fact that burn rate is pressure dependent. No pressure creates a kind of fizzle that spreads out the substance faster than the flame can spread, resulting in no significant explosion.
Minor things could have serious negative effects at least on existing firearms not designed for such environments - like subzero temperatures on lubricants, packing of powder in a vacuum etc.
Theoretically, you would have marvelous range with no fricition or low gravity.
