The PSP has a hibernate mode, which saves your exact position when you turn off the console. When you turn on the console, you will be in the exact same position, whether in the PSP system menus, or actually playing a game.
If your battery dies, all unsaved data will be lost. If the green power LED on your PSP is flashing, it is best to save your game, as this indicates low battery.
To hibernate your PSP, simply push the power switch up for no more than 1 second. To turn off your PSP completely, hold the power switch up for 3 seconds.