It's well seen throughout the series that there is a “curse” on the DADA teacher position, that no one can hold that post for more than a year. But it seems like Quirrell has been around for longer than a year. What's up with that?
Quirrell was the Muggle Studies teacher, prior to his travels. Once he met Voldemort in Albania, he returned (for Harry's first year) as Defense Against the Dark Arts teacher, (failing to last more than a year).
For confirmation, see this JK Rowling interview at The Leaky Cauldron:
Has the jinx on the DADA teaching post at hogwarts been lifted?
Yes, at last! Incidentally, I know some have asked about Quirrell with regard to this question.
He was teaching at Hogwarts for more than a year, but NOT in the post of DADA teacher. He was previously Muggle Studies professor.
As I explained in this answer, Quirrel was apparently the Defense Against the Dark Arts teacher for more than a year. As for how it fits with the curse on the position, I explained there that there probably was no such curse.