With the exception of downloadable titles, the upcoming PS Vita will not be compatible with PSP games due to its lack of a UMD drive. However, Sony plans to solve the problem of backwards compatibility with the UMD Passport application that will allow PSP owners to re-download games they already own.
Not for free, mind you. No no. With a discount.
The Passport system will be rolling out in Japan ahead of the Vita's Christmas launch, and it works by downloading the app onto your PSP, booting it up and inserting the UMDs you own. The application will take care of the rest by syncing the details with your PSN account. Note that each UMD can only be registered with one account - so no cheating I'm afraid. From there, you can merrily download it at a discounted price level... or, you know, just keep playing it on your PSP that you'll already need to own in order to use the app in the first place. Just sayin'.