Joining its announcement of the UMD Passport program (details here), Sony has shared specifics on what players can expect when playing a PSP game on PlayStation Vita.
Boot up a PSP game on Vita, and you'll see the following screens:
The screen to the left is the LiveArea for the PSP game. If the game has a digital manual, you'll see an icon for the manual here.
To the right are some play options. You can toggle bilinear filtering, change settings for the right analogue stick and switch the color space to match the color of the PSP. For games that use the system's camera, you can switch between the front and back camera.
You'll be able to transfer your PSP saves to your Vita. The transfer can be done through the PS Vita Content Management program on your PS3 or PC. In the case of a PC, you'll need to install a program called "Content Management Assistant for PlayStation." This will be released later via the PlayStation.jp site. (Sony didn't say this, but the PS3 content management program will presumably be added to the system via firmware update).
If you have a UMD copy of a game, you'll need to use Sony's UMD Passport program to transfer it to the Vita (assuming it's supported by the program). For download versions, you can download the game directly from PS Store, or download it to your PS3 and transfer it from there to your Vita.