With a Custom Firmware, you can make backups of your PSP games (in .iso/.cso/.dax format) and then play those backups if your UMD disc gets damaged/lost/whatever. You can even compress your backups!
How to make a backup of your UMD:
1) Put your UMD disc into your PSP.
2) In XMB, launch VSH Menu by pressing the Select button of your PSP. VSH Menu is in both PRO and (L)ME CFW's.
3) In VSH Menu, navigate to USB Device (usually third option) and change it to "UMD Disc". With this settings, you'll see the content of the UMD Disc instead of memory card when you connect your PSP to computer.
4) Connect your PSP to your computer like you always do and start USB Connection.
5) Copy the .iso file to the computer. Don't forget to change USB Device in VSH Menu back to "Memory Stick".
Alternatively you can use this Homebrew-Tool: UMDumper v2.4
How to play a .iso backup:
1) Connect your PSP to your computer.
2) Go to the root of your memory stick. There should be a folder called "ISO". If it's not there, create it.
3) Copy your .iso backup to the ISO folder.
4) Disconnect/unplug your PSP.
5) You should see the .iso backup, that you just copied to the PSP, in the "Game" category of XMB.
You can also play PS1 backups, with the built-in PSX emulator:
2) (Assuming you have a PS1 backup) Then you have to convert that backup so it can be played on the PSP. Here's how to do it.
3) After converting, you should have a folder (called after your game) with EBOOT.PBP file in it.
4) Connect your PSP to your computer. Copy the folder with EBOOT.PBP to root/PSP/Game/[here].
5) After disconnecting, the game should be in the "Game" of XMB.
The last backups you can play on your PSP with CFW are games released on "old" systems, like NES, SNES, N64 (to an extent), etc. You need Emulators for that, and more information about them is in the Emulators section.
