The PS Vita is a pretty phenomenal piece of hardware that's technically capable of being backward-compatible with the entire PSP library. Alas, at this point, the list of PSP games you can download onto your PS Vita is limited. Sure, there are plenty of great older games (and lots of great brand-new ones, too), but there are many missing titles. Browse the list for yourself.
Sony says the list of backward-compatible games will continue to grow, but why wait? There are actually plenty of other PSP games you can play on your PS Vita right now, but you'll need a PS3 to make it happen. They may not be listed on Sony's backward-compatibility list, but they certainly work.
So, here's the trick, which has been discussed on many message boards: while you can browse the PS Store to see what older downloadable PSP content is available for purchase or redownload, you're not getting the full picture of the whole PSP catalog. Instead, browse your download list via Transaction Management on your PSN account on the PlayStation 3, and download any PSP games. The PS3 will store PSP games and allow a PSP--or, a Vita--to sync the content.
Connect your PS Vita to the PS3 via USB cable (half of the AC charging kit that came with the system) and launch the Content Manager app. Select the "Connect to a PS3 System" option, and select "Copy from PS3 to Vita." You can browse all PSP-related save content. At this point, any PSP games that can be transferred show up in a list. Select them, copy the files, and you'll have your PSP games on your Vita.
So far, I've done this to transfer two of my favorite games: Echochrome and Flow. According to message boards, many more games are transferable, but the only officially supported games are those that show up in the PS Store, and are listed in the official Sony blog post above. Of course, there's a good chance that these other games will officially appear in the PS Store sooner or later. And, sorry, UMD owners: this still doesn't help you at all.
Early PS Vita adopters (the Vita officially launches tomorrow), explore away. Any successfully transferred games you'd like to share? Let us know in the comments.