In the comics, he was once a human named Arthur Douglas, whose spirit was placed inside an other powerful body, and then Drax the destroyer was created.
Unlike the movie version, when he was human, his family was killed by Thanos, instead of Ronan.
In the early comics when he was introduced in the Marvel universe, he had the ability to fly and could also shoot energy blasts.
But when he became a member of the Guardians of Galaxy (relaunched series), he lost the flight ability and the power of energy blasts.
Now he just has great strength and strong senses and likes to use knives in battles, even though he's not that much of an expert in fighting with them. But he once passed through Thanos's force field (I don't remember what issue it was), so I guess he has some great superhuman strength !
I read somewhere that he was made by Gods to kill 'the mad titan Thanos', that's why he's called the Destroyer. I'm not sure if he's same in the movie version.