Update: The new PSP-3000 cannot currently be downgraded, and PSP-2000 models with a TA-088 v3 motherboard cannot be downgraded either, ALL other PSP models can be downgraded, (including the newest PSP-2000 which has TA-090)

If you have PSP Firmware Version 3.51 or higher, you CAN downgrade, however you will need a Pandora's battery and a "special" memory stick.  To obtain this battery you must either purchase it from a seller, make one yourself by using a PSP that is capable of running homebrew (a PSP with version 1.5 or custom firmware), or hardmod an official battery.

Pandora's Battery can downgrade ANY compatible PSP regardless of the firmware installed, and can even recover "bricked" PSPs.

This guide answers these questions about the PSP (PlayStation Portable):

• What is so special about firmware versions 1.5, 3.03, 3.50, 3.71 M33, and 5.00 M33?  What is OE-A and M33?  What is custom firmware?
  
• Why would I want to downgrade a perfectly functional PSP?  Doesn't that mean I would lose cool features (web browser, PS1 emulator, etc)?
  

Here is the simple answer: the reason to downgrade your PSP is to "upgrade" it!

That's right, you will get more out of your PSP by downgrading (to v1.5), because you will then be able to install custom firmware (v 5.00 M33), and have full, unrestricted access to your PSP, instead of Sony's limitations.

You will be able to play homebrew (software by someone other than Sony) games and applications, you will be able to play YOUR PlayStation 1 games on YOUR PSP (nearly every PSX game is compatible), and of course emulators (I won't go into it, but everything between the Amiga, the N64, and even the PSP itself can be emulated, also you SHOULD own the games you're emulating).

Version 1.5 firmware will allow you to run all homebrew, but you will lose your web browser and everything else, and your retail UMD games will be unplayable and require you to update your firmware or run a homebrew program called devhook.  That's why I recommend installing custom firmware 5.00 M33, which has all the capabilities of the official 5.00 firmware, plus the ability to run homebrew, and play every retail game currently available.

Now that you know about the benefits, you probably want to know how to downgrade/upgrade.  I won't give you instructions on how to downgrade, as there are quite a few steps, and tons of tutorials on the internet, but I'll tell you the basics.

• Find out your PSP firmware version by turning on your PSP and going to Settings -> System Settings -> System Information, then read the number to the right of "System Software", this is your firmware version
• If your version is 3.51 or higher, you MUST use Pandora's Battery to downgrade to 1.5
• If your version is 1.5, you can already run homebrew, and upgrade to custom firmware
• If your version is 1.51, 1.52, 2.0, you can downgrade by upgrading to 2.0, then downgrading to 1.5 with NO additional hardware
  
• If your version is 2.50-3.50, you can downgrade by upgrading to 3.50, then USE Lumines to downgrade to 1.5
  
• You can also use GTA: Liberty City Stories (2.0 edition) to downgrade any version below or equal to 3.03

• When you change your firmware (by downgrading/upgrading) you risk "bricking" your PSP (corrupting the internal memory permanently), ALWAYS plug in your PSP and have a fully charged battery when changing firmware
• READ and FOLLOW the downgrading instructions exactly, DON'T SKIP STEPS UNLESS YOU'RE A PRO
• DON'T USE FAKE MEMORY STICKS, they go bad and can corrupt during an upgrade/downgrade
  
• MAKE SURE the software you are downloading is the correct version, WATCH OUT for VIRUSES that can destroy your PSP

Links:
hxxp://en.wikipedia.org/wiki/PlayStation_Portable_homebrew
hxxp://pspupdates.qj.net/From-Dark-AleX-with-love-PSP-CFW-5-00-M33-released/pg/49/aid/125038

That's all for now, I don't want this guide to become huge and difficult to read, thanks for looking at this guide!