As far as I can tell, it is difficult to tell the full difference between Satan and Lucifer. Perhaps there is validity to the Islamic belief that the devil was a Djinn, and while Lucifer was a fallen Angel, Satan was actually a Djinn, and demonic in nature.
When Allah commanded all of the angels to bow down before Adam, Iblis, full of hubris and jealousy, refused to obey God's command (he could do so because, as a jinn, he had free will), seeing Adam as being inferior in creation due to his being created from clay as compared to him (created of fire).
"I am better than he: Thou didst create me from fire, and him from clay." Qur'an 7:11-12
It was after this that the title of "Shaitan" was given, which can be roughly translated as "Enemy," "Rebel," "Evil" or "Devil".
Shaitan then claims that if the punishment for his act of disobedience is to be delayed until the Day of Judgment, that he will divert many of Adam's own descendants from the straight path during his period of respite. God accepts the claims of Iblis and guarantees recompense to Iblis and his followers in the form of Hellfire.
In order to test mankind and jinn alike, Allah allowed Iblis to roam the earth to attempt to convert others away from his path. He was sent to earth along with Adam and Eve, after eventually luring them into eating the fruit from the forbidden tree.