I understand that the possessor is in the genitive case but what is the difference between
машина девушки and
машина у девушки and
у девушки есть машина and
я вожу машину девушки and
я вожу машину у девушки?
I would assume that "девушка" in these sentences stands for someone's girlfriend though it could be some other girl.
машина девушки is most simple – "my girlfriend's car"
машина у девушки is usually used as a part of a sentence. Like "а что за машина у девушки?" — "What kind of car does your girlfriend have?" Or "моя машина сейчас у девушки" — "My car is currently at my girlfriend's place"/"I gave my car to my girlfriend for now"
у девушки есть машина – "my girlfriend has a car"
я вожу машину девушки - "I drive the my girlfriend's car"
я вожу машину у девушки – This doesn't sound naturally. I'd translate it as "I drive a car at my girlfriend's place" or something like that.
The combinations of words "машина" and "девушка" in numbers 1 to 4 express approximately the same meaning -- "car [which] belongs to girl [and I drive it]". Number 5, however, might sooner mean "I drive the car [in the company/firm/organisation] belonging to the girl". It would hardly relate to the girl's possession of the car itself.