Really? Firmware is the proprietary software that Sony uses to run everything on the PSP, it’s sort of like an OS but it goes a bit deeper (firmware is in stalled to your PSP’s flash memory thus it is booted anytime the PSP is started).
A custom firmware (CFW) is a hacked version of Sony’s software that removes restrictions to allow you to use homebrew (a program written by an author that is not authorized by Sony to produce for the PSP) or to play backed up games (games that are copied from the original disc and are read from the Memory Stick) or to tweak the performance of your PSP.
A kernel is the central part to a computing device, it is a software that transmits and translates information between memory, CPU, higher level applications, and hardware interfaces.
The kernel is different for each firmware. Older PSP firmware kernels (1.5 especially) allowed certain exploits to be used in order to execute unsigned code (homebrew). With the newer firmwares Sony eliminated the ability to use these exploits by changing the kernel. In order to play homebrew on a PSP for a long time the only option was to use a PSP with original 1.5 version firmware. Because of this a large majority of homebrew programs were based on the 1.5 kernel. Later on when custom firmwares started becoming popular older kernels were being incorporated into the custom firmwares to allow people to still use the old homebrew that relied on the 1.5 kernel.
Sony combats custom firmware advocates by releasing newer firmware and making new games require this new official firmware. Gran Turismo was released encrypted (a way of making information unreadable except by firmwares that know the code). The 6.00 firmware allows the PSP to interpret the information in the game; however, there is an alternative which is to patch the game by decrypting it so it can be used by the 5.50 firmware.
Only one reliable 5.50 custom firmware exists that I know of. M33 is usually the homebrew team I go to when I want a reliable custom firmware, it is headed by a guy that has been making reliable CFW’s for years. Because M33 is not making a 5.50 or 6.00 CFW the 5.50 GEN custom firmware is all there is right now. 5.50 GEN allows a modded PSP to run decrypted 6.00 games and some homebrew that was made for later kernel versions; however, this CFW doesn’t have the option to install the 1.5 kernel so the old homebrew doesn’t work on it.
I have to ask: How did you get this far without knowing what any of this means? Did someone just give you a modded PSP? Surely you’ve looked into some of this before trying to install and play games on your PSP?