Well, those of you who are new to homebrew will need to know how to install it, so here's my guide. Please refer to your firmware revision below to find out YOUR needed method.

Firmware v1.0

This is the easiest firmware to install homebrew on. Steps are as follows:

1.Download your homebrew, and "Un-zip" it. (You will need an unziping program to do this. We recomment WinRAR). The Best place to get homebrew is either
Here, which is owned by "Fanjita", the creator of the GTA Eloader, or HERE, which belongs to QJ.net.

2.There should be one or two folders inside the file. The First should be "Homebrew" (Substitute the word homebrew for the homebrew's name), and "Homebrew%".
For firmware 1.0, you don't need the "Homebrew%" folder. Simply drag-and-drop the other folder into the following directory via USB*:
X:\PSP\GAME\ (Replace "X:" with the drive letter of your PSP, and remember that the folder named "PSP" is INSIDE the drive itself.)

3.Disconnect the device, and in the PSP's main menu, go to "Game", and then "Memory stick". Your homebrew should be listed, just press [X] on the icon to launch.

* To put your PSP in USB mode, go to "SETTINGS>USB MODE".

v1.50

Download THIS
Before, you had to use the "Homebrew%" folder, too. Now, you run this, and you don't have to worry.

To be able to use this, though, you need to drag-and-drop both the "NOKXPLOIT" and the "NOKXPLOIT%" folder into X:\PSP\GAME\ (Replace "X:" with the drive letter of your PSP, and remember that the folder named "PSP" is INSIDE the drive itself) Via USB, and keep them there PERMANENTLY.

Then, just follow the steps in "Firmware v1.0"

v1.51 or v1.52

There has never been a way to launch homebrew on these firmware's. To be able to, update to a higher version (Preferably 2.00), and if you so wish, downgrade to v1.50.

v2.00, 2.01, 2.50, 2.60: GTA Eloader

 The TIFF eloader is much better for these versions. If you want to use that, see the below section. Anyway, the following is for referance:

Thanks to "Fanjita" and "Ditlew", well known PSP hackers and coders, we have the ability to launch homebrew on these firmware revisions. After completing the following steps, Boot up "GTA: Liberty City Storys", just load up the "Eloader" GameSave, and go!

Follow these steps:

- Download the lastest version of Eloader from
HERE
and choose the latest version available on the page.

1.Install using either the Windows installer, or copy the files manually on Mac, Linux, or other platforms from the downloadable archive. Make sure you install the GTA version.

2.Follow Steps 1 and 2 in the "Firmware v1.0 section"above.

3a. (optional) If you want the Eloader to automatically load when you start GTA, leave it as it is, otherwise, do the following:
i) Go to "X:\PSP\SAVEDATA\" (X is the letter of your PSP), and find another GTA game-save. You can tell them by the icons.
ii) Either, copy the other GTA game-save folder to your PC, delete the original and copy back. Or, Create a new .RAR archive, add the other GTA game-save to it, delete the original folder, and copy back. Delete the .RAR archive.

3b. To load your Homebrew, boot up GTA, and when you're character appears, and everything is loaded, press START. Click LOAD, and then select the Eloader Game-save. Confirm it.

4. After a few seconds, a menu shold appear, with a list of installed homebrew. Select your desired application, and press [O] or [X]. Done!

Extra: Remember that not all hombrew is compatible with the Eloader.  Also, if you would like to return to Eloader while homebrew is running, hold down [Left trigger] + [Right Trigger] + [START] for five seconds. This does not work with all homebrew.

NB:

#1There has is a "Downgrader" released for these versions, so please read our "downgrading" guide.

#2 There have been reports that newer GTA:LCS disks block the use of Eloader and other Homebrew using this exploit. If you want homebrew, buy PRE-OWNED versions!

v2.00 - v2.60: TIFF Eloader

 The 2nd TIFF exploit that fully works on firmware versions v2.00 to v2.71 has brought about a new version of the Eloader, allowing us to use homebrew without the need for GTA. 

It's easy to set up and use. 

1. Download the Eloader installer from here: [LINK]. Use the installer for Windows preferably. If you do, make sure you select the TIFF version from the installer.

 If you're not a Windows user, just read the readme.txt file. The basic lines of it are extract the files to the PSP. Simple.

2. Install the homebrew you want using Step Two from the v1.0 section.

3. Once installed, unplug the PSP from USB and go to the "Photo" menu. Click on the folder that says "eLoader" on it, and the PSP will freeze. It'll load up the eLoader. If it doesn't, try again, and make sure everything is installed correctly.

4. Every thing you need to know is found in the help section of the eLoader. To get to it, just press [SELECT] while in the eLoader menu.

v2.70, 2.71, 2.80: TIFF applications

The TIFF eloader works on these versions, but general applications have been released using the Noobz  2.71-2.80 SDK. These include Snake, Pong, TRON, and Tetris. Note that these will also work on any other Firmware that is capable of using the exploit. (IE. 2.00-2.80).

You can find the fantastic eloader here: [LINK] To install homebrew, follow the tutorial in the section above.

Tron can be found here: [LINK]. Credit goes to the Noobz team for the SDK, and "Elad_THC" for the application itself.

Pong can be found here: [LINK]. Credit goes to the Noobz team for the SDK,  "Robert A." (From PSP3D), and also "Percival".

Tetris can be found here: [LINK]. Credit goes to the Noobz team for the SDK, and also for the application itself.

Snake can be found here: [LINK]. Credit goes to the Noobz team for the SDK, "Youresam" for the port, and "Shine" for the original game.

The SDK can be found here:  [LINK]. Make sure you read the read-me.

Dark_AleX's 2.71 SE custom firmware

This supports v1.00 and v1.50 homebrew, so follow those tutorials. HOWEVER, in SE-A (The first version), all v1.50 homebrew must go in the 'PSP\GAME150' folder, as v2.71 homebrew goes in the GAME folder. 

In later versions, you can put them in the GAME folder, but only if the GAME folder is set to run with the v1.50 kernel. You can enable/disable this in recovery mode, by holding [R] at boot, in the '>Configuration' area. 

ISO's appear in the GAME>Memory stick menu in the XMB. Make sure they're placed in 'ISO' in the root of your memory stick.

v2.81, v2.82, v3.00, v3.01, v3.02, v3.03

Firmware v2.81 was released to block the TIFF exploit, so no homebrew can be run on that version or later. Sorry.