The only way to run homebrew on the PSVita is via PSP Mode exploits. To use those, you need to purchase (for real money) a given exploitable game on PSN and then install the exploit files according to the simple instructions that come with the exploit package.
The best way to find out about the latest exploitable games is to register at wololo.net forums and await further developments - if you frequent the forums, you will be notified about the next exploitable game before "the rest of the world" and before Sony can take the game off the store.
As of today, there are no exploitable games announced for the 2.12 firmware, however new exploits are in the works.
As you are a PSP owner, I'll give you a word of advice - don't bother. Using an exploit like this pretty much equals not being able to update the system anymore (as you will lose the capacity to launch homebrew the moment you do update) so you can practically forget about using online features or taking advantage of the quite stellar PSPlus program. It's far more advisable to launch PSP homebrew and backups on your PSP - it's just less troublesome in my opinion. Of course if you choose to do otherwise, go on ahead.