The last official Sony firmware for PSP is 6.61 (released 2015).
Custom Firmware (CFW) like 6.61 PRO-C2 or 6.61 Infinity adds homebrew, ISO loading, plugins, and perma-patch. CFW installation:
For most users, 6.61 OFW + signed homebrew (via HEN) is safer. CFW for enthusiasts only.