As of this moment there are no fully capable PSX emulators for the PSP. There is a PSX emulator out there in development but in the meantime we can only wait.
If you want to play emulators or homebrew on your PSP, the longer you wait the worse off you'll be. If you're in the US or Canada you need a PSP with firmware version 1.50 (not 1.51, not 1.52, 1.50 period end of sentence.) which are quickly becoming harder to find; you may only be able to find a 1.5 PSP at a shop where they sell used systems.
If you're in Europe, you're outta luck, only 2.0 PSPs are available across the pond, unless you find somewhere that will import PSPs.
Also if you want to play emulators, games are coming out now that REQUIRE the 1.51+ firmware (Namco Battle Museum and Burnout Legends notably).
So in the end it's a tradeoff. If you want to play your NES, Atari, Genesis, etc. games on your PSP you may have to do a little bit of work to get it (it's still not an easy plug-and-play type thing either to put homebrew on your PSP), and if you want to play any of the new games coming out, you'll be out of luck on homebrew. The 2.0 firmware is great though, with the web browsing capabilities and background changing abilities too. Oh, and it's only a matter of time until 2.0 is hacked (there's already a small exploit for it right now, it shouldn't be long until executable code can be... executed.)
If/when you get a PSP feel free to contact me, as tutorials to PSP homebrew are a little thin. But here are some good sites to get started:
www.psp-hacks.com
www.pspupdates.com