This is a very good question...Dumbledore has to trust Snape to some degree, because he hired Snape to teach at Hogwarts. (But he doesn't let him teach Defense Against the Dark Arts; maybe he thinks that it would be too tempting for him, being amongst the dark arts again? Or else Dumbledore trusts him, but his trust is limited in some areas.) Not only did Dumbledore hire Snape as a teacher, but Snape is also the head of Slytherin House, in charge of 1/4 of the students in Hogwarts.
Snape was into the dark arts ever since he was young, so I guess he naturally went over to Voldemort's side when Voldemort gained power. We don't know yet what it is that caused Snape to stop following Voldemort...I hope JKR tells us sometime. :-) Maybe Voldemort killed his family, or someone he loved (Lily Potter??), or maybe Snape just realized that what they were doing was wrong. Dumbledore obviously knows what happened---actually, he seems to know many things that nobody else knows. It's a great character to have in a book; whenever Rowling wants to introduce a major plot twist (like the prophecy), she can just use Dumbledore to explain all about it because he knows everything. :-)
So, I think that Dumbledore trusts Snape because he knows why Snape joined the Order of the Phoenix in the first place, and we do not. I really, really, really, really hope that Snape doesn't join the dark side again in books 6 or 7...I want him to be a good guy.
Do you think that he ultimately is a good guy, anyway? He is very nasty and unfair to some of the students, but he keeps them safe. Heck, he even saved Harry's life in the first book. Harry
, the student he hated the most. If Snape cared enough about Harry to look out for him in the first book, doesn't that show that he's really a good guy at heart? :-)