define: emulate - Google Search
An emulator is a program that sets up an environment that allows programs (games) to run on a system other than the one they were written for without modifying the code.
In terms of games on the PSP, emulators allow you to play games from other consoles (SNES, Genesis, GBA etc) on your PSP.
Emulators are unofficial programs so they are considered homebrew.
There is an official PS1 emulator for the PSP by sony also.