Now, back to matters at hand.
Propulsion in space is actually supremely simple. To accelerate in one direction, you throw something in the opposite direction. Matter/antimatter reactions, plasma or ion drives, chemical or nuclear rockets, it all works on that same basic principle of action and reaction. For all intents and purposes, you could load up an orbiting spaceship with a large cache of heavy rocks, and travel to mars simply by continually lobbing them out the back. Assuming you lived that long and had enough rocks, of course, neither of which would happen.
So there's no requirement for oxygen, as there's no requirement for an explosion. Energy, of course, is
needed, but contrary to what you say, near space is actually chock full of energy in the form of sunlight. And in deep space far away from any stars, many of the methods that work here on earth work in space as well; nuclear reactors are not limited by availability of oxygen and so are a reasonable choice, as are radioisotope generators as evidenced by the Voyager probes.
For practical reasons though (namely, having to overcome gravity), all spacecraft lifting off from Earth still use chemical rockets, which do in fact burn perfectly well in space. Sure, there's no oxygen there, but they carry oxidizers as part of their fuel mix which gives the same net result. N2
O, better known as nitrous, is an example of this kind of stuff. What happens when it's introduced into a car engine is it adds oxygen to the combustion, and in space it's no different. Only instead of improving a reaction, it simply allows it to happen.
Now, knowing this, it's not a big leap to see that a firearm using bullets that carry an oxidizer in addition to the explosive would work perfectly well in space. So that's one weapon. Missiles and rockets will also work, as long as they're once again designed to carry oxidizers - Or you could stick an ion drive powered by a radioisotope generator on them, and skip the oxidizers altogether. Rail and coilguns, as Aldo already mentioned, just require energy, and so will work fine. Then there's lasers... see a trend here? All these examples are doable and have been demonstrated with today's technology, so weapons in space is not science fiction.
The one thing you have to be careful of when using weapons in space is what I discussed in the first paragraph. Throw something in one direction, and you'll accelerate in the other. And unlike on earth, there's no air friction to stop you. So be careful about firing that chaingun too much or you might just end up deorbiting yourself