Methods overview
Custom firmware on the PSP
Custom firmware (CFW) unlocks the PlayStation Portable, letting you run homebrew applications, emulators, backups of your own games, and plugins that extend the system. The good news for PSP owners is that nearly every model can run CFW, and the process is far simpler today than it was during the console's commercial life.
The standard software route follows three stages. First you update the system software to 6.61, the final official firmware released by Sony. Next you run a CFW installer such as ARK-4 or 6.61 PRO, which boots you into a custom environment. Finally you install 6.61 Infinity, which patches the system so the CFW loads automatically on every boot instead of needing to be re-launched.
Which models can do this?
All retail models run the software method: the PSP-1000 (Phat), PSP-2000 and 3000 (Slim), and the PSP Go (N1000). The E1000 (Street) is also supported. The only hardware-specific route is Pandora's Battery, covered separately.
The hardware alternative
A second, older path uses a Pandora's Battery and a Magic Memory Stick. This is a hardware recovery method aimed mainly at the PSP-1000. It is the go-to option for unbricking a dead console or installing older firmware, but for most healthy systems the software route is faster and lower-risk.
Read the methods comparison and the temporary-versus-permanent explainer before choosing your path.