You charge the PSP battery inside the PSP using the Official AC Adapter.
Standalone chargers exist (e.g., Datel Max Power, Generic 2-bay). Useful for swapping spares. Ensure 3.7V/4.2V Li-Ion compatibility.
Warning: Do not use generic Li-Po chargers (Wrong voltage curve). Use PSP-specific or 3.7V Li-Ion CC/CV 0.5C-1C (900mA-1.8A).