Sony has announced that next week they will be releasing the new Vita firmware 1.80 on August 28th, and it will add support for PSOne games. The Sony PSN store already has loads of PSOne games, so it will instantly allow these to work.. but Sony has also announced it will be making at least 100 PSOne games available for Vita users.

More detail about the v1.8 firmware update:
Sony announced today that PSone support will be added to Vita later this month, but that's not all the upcoming firmware update will bring.

Firmware update 1.80 - the update that'll include PSOne Classics support - will also add the Wii U-like Cross Controller mode, a feature for controlling PS3 games with the Vita, as demonstrated with LittleBigPlanet 2 during Sony's Gamescon press conference today.

Playlists will be added to the system's music player, along with new playback speed control and repeat play options in the video player.
Rear touch pad support will be added to the web browser for zoom and scrolling and, at long bloody last, button controls will be enabled on the console's home screen and standard applications such as Videos and Music.

Photos will support the MPO format, you'll be able to take photos to add as attachments when writing Group Messages, and maps will also get some small usability improvements.

The update will be released on August 28. For the full nitty-gritty details, head through to the US PS Blog.
Quoted from CVG

And more about the PSOne support:
Sony has said that over 100 PSone games will be available to download and play on Vita next week.

The PlayStation Store is already brimming with PlayStation titles of yesteryear including Metal Gear Solid, Resident Evil and numerous Final Fantasy games. Although it didn't mention any specifics, we expect Sony is planning to add new classics periodically. "Looking to downloadable products, Jim Ryan revealed that from August 28th, over 100 PS One classics including Final Fantasy VII, Metal Gear Solid, and Resident Evil 2 will be available to download for PS Vita," reads a press release.

The much-requested PSone support for Vita was revealed during Sony's GamesCom conference earlier this month.

The functionality will arrive with Vita firmware 1.80 on August 28. The update will also enable the Wii U-like Cross Controller mode, which allows players to control compatible PS3 titles with Vita, as well as music playlists, rear touch pad support for the web browser, and additional home screen button controls.
Quoted from CVG