Homebrew enabler (commonly shortened to HEN) is an application that makes use of exploits on the 2.00-3.50 firmwares, and games, that allow homebrew to be launched and run from the PSP, as long as HEN is active. Often bundled with HEN would also be a downgrader allowing users to downgrade their PSP back to 1.50.
2.00 - 2.50 HEN
The first homebrew-enablers came out when an exploit was found in Grand Theft Auto: Liberty City Stories UMDs that allowed execution of homebrew. With this exploit found, HEN was developed and released for 2.XX firmwares. Sony released newer firmware, and patched up the exploit found on the discs. Also, the UMDs are getting patched and upgraded to "Greatest Hits" status, in which firmware 2.60 was required, instead of the normal 2.00.
2.00 - 2.71 HEN
After that another exploit was found, the TIFF exploit. A new version of HEN came based on the TIFF exploit, that could work on any FW up to 2.71.
3.00 - 3.03 HEN
On 3.03 firmware, the Grand Theft Auto: LCS UMD exploit (now known as the Goofy exploit) was still able to be launched. In nearly the same way as the 2.xx enablers, this allowed any PSP to use homebrew or downgrade. Sony quickly patched the FW, rendering the Goofy exploit quickly useless.
From 3.30 onwards, Sony started to have a check included on their firmwares. This check will check the IDStorage Keys of a PSP, and decide whether or not the PSP would be allowed to upgrade. If the IDstorage keys were in any way corrupted, missing, damaged, or not of normal values, the PSP will encounter an error.
This provided a disadvantage for users who had previously downgraded their firmwares, and then wanted to re-upgrade past 3.11. However, Noobz developed a downgrader that makes use of the Illuminati exploit, and allows users to downgrade back to 1.50, and be able to run an IDStorage Key Analyzer in which can patch keys to prevent upgrading, or restore the default ones so upgrading can be allowed again. Also, a user can be able to upgrade back to custom firmware if he/she so please.
3.00 - 3.50 HEN
However, another exploit (the Illuminati exploit) was found in the game Lumines. The 3.50 HEN is based on this. Sony patched this exploit in both the new FWs, as well as on the Lumines UMDs. Further Developments
Ever since the creation of Pandora Battery, there hasn't been much need to further develop further homebrew-enabler releases. Mainly, for two reasons: The Illuminati exploit has been patched in firmware 3.51, and thus ended exploits via Lumines, and that Pandora Battery can downgrade any PSP regardless of motherboard, region, or firmware.