Firmware is the software built into the PSP. It controls what you can runs, and contains the interface for the PSP to connect to your computer or do anything else the firmware allows.
For the PSP there are many different versions out right now, and their functions and options are as follows:
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Firmware Version 1.00:
Allows unsigned code, or "HomeBrew" to run. Homebrew is defined as any type of program that was written by an average user (Not Sony or an affiliate) to run on the PSP.
This version does not include an official browser and was only ever found on Japanese PSPs.
Firmware Version 1.50:
Allows unsigned code ("Homebrew") to run using an explot in the security of the firmware.
This version was found on the original release PSPs in the Americas. IT would not oroginally allow homebrew. It contains no official browser.
Firmware Version 1.51:
Does not allow unsigned code. Other than that, it's the same as the 1.50 firmware.
Firmware Version 1.52:
Does not allow unsigned code. Other than that, it's the same as the 1.50, with only an option for playing music UMDs added in the music menu.
Firmware Version 2.00:
This version was a fairly major update. It added the option for an official Sony browser, the ability to use your own backgrounds with ease, .AVC video support, and some more encryption settings for wifi. It didn't originally allow homebrew, but using an overflow exploit, it can now run SOME homebrew games. Also, using the exploit, you may downgrade to 1.50, which will allow you to play all the homebrew you want.
Firmware Version 2.01:
This version was the same as 2.00, except the security flaw that allowed for homebrew was fixed.
Firmware Version 2.50:
This is basically a version 2.01 for rich people. It's the same as 2.01 except for the fact that if you buy a $300 US Location Free Player Dock, you can play TV from wifi hot spots anywhere around the world.
Firmware Version 2.60:
This is firmware version 2.50 with a few added features. You may not play .wma files (Windows Media Audio) on the PSP, but this function must be activated through the internet. Also, you can now play 480x272 video on the PSP. RSS feeds are now available (see Here for mote info on RSS and WMA)
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Notice!
Firmware Versions 1.51, 1.52, 2.01, 2.50 and 2.60 WILL NOT RUN HOMEBREW. (not including 2.01 tetris)
Firmware versions 1.51, and 1.52 may run homebrew by upgrading to 2.00, then either running minimal homebrew, or downgrading to 1.50.
You CAN NOT downgrade from 2.01, 2.50, or 2.60. Period.
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If you need to know which firmware you have, go into your PSP's Home menu, go to Settings -> System Settings -> System Information
The version will be listed under System Software: x.xx
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To upgrade your PSP's firmware Either use network update (This will give you the most recent firmware version) or download the eboot.pbp on your computer and put it on your memory stick via USB or Memory Stick reader. The ebbot.pbp file should go in x:/PSP/GAME/UPDATE , where the folder names are all in capitals.
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Now that everyone knows please do not ask about firmware anymore.