Emulators on PSP
RetroArch for PSP has cores for GB, GBC, GBA, NES, MSX, PCE and PCECD. Also Cave Story- and Doom-engines. The Gambattle core for GB/GBC works at full speed. The PC Engine emulator is also very good.
PlayStation / PS1
See PSP Eboots.
Game Boy Advance
TempGBA4PSP is the most recent GBA emulator for the PSP, and has superior compatibility with most problematic games (Golden Sun, for example).
Game Boy / Color
GeMP and RIN have some incompatibilities with later PSP models. GeMP is incompatible with GameCategories.
Famicom (NES) / Famicom Disc System 
AoEX is based on NesterJ 1.12 Plus 0.61 RM, so it includes features like rewind mode, cheat codes support, rotated/mirrored screen, sepia palette, support to rare mappers (the pirate bootleg FF7 works on it), etc, but its compatibility is inferior to 1.13 beta2.
Super Famicom (SNES)
SNES emulation isn't very good to begin with, but this is the fastest and most compatible emulator available right now. Note that the mecm version should be used if on an actual PSP; if on a Vita, use the cm version.
Most games are unplayable, but there's a small amount of them that work really well with the right settings (Quest 64, for example).
Genesis / Sega CD 
Master System / Game Gear 
MasterBoy 2.10 (also compatible with GB and GBC)
PC Engine / CD 
Neo-Geo Pocket / Color
WonderSwan / Color 
e(mulator) v082f (multi-emulator)
NJEMU (CPS1, CPS2, NeoGeo MVS, NeoGeo CD)
Other Consoles (retro)
Atari 130 / 800 / 5200
Commodore Plus / 4
MSX / MSX2 / MSX2+ / MSX Turbo-R
Sinclair ZX Spectrum
TRS 80 / Tandy Color Computer