Sony released a long list of PSP games that will definitely be playable on the PS Vita. You might expect that every game downloadable through PSN should just work automatically, but that doesn't seem to be the case.
The initial list features 275 PSP games and PS Minis that Sony guarantees will work on the Vita. Sony has said that they're working to make more games compatible, so the list will grow as time goes on. Check out the full list here.
Don't go off deleting your games just because they're not on the list this second. Joystiq has already confirmed at least two games not on that list that should work on the Vita right away; Valkyria Chronicles 2 and PixelJunk Monsters Deluxe. Though it's not explicitly stated, Sony's list is most likely limited to games they're sure will work perfectly on the new platform.
Sony has also posted a guide to transferring games to your new Vita either via PSN or your PS3 if you've downloaded them there. There's nothing especially complicated about the process, but you should definitely take a look if you run into trouble.
[Via Playstation Blog]