What's the difference between a bomb, a missile, and a rocket?

A Bomb is unpowered (though they can be guided)
A Rocket is powered, but unguided.
A Missile is powered and guided.

There are exceptions to these definitions in that they only hold true for modern weapons of war.

An arrow is also termed a missile, though it is unpowered and unguided (after it has left the bow).

Space borne rockets are both powered and guided (unless they are ICBMs in which case they are modern weapons of war and termed missiles).