There are a few makers, Unconventional Airsoft has one (Trigger Master), then there's the Extreme Fire
ones that supposedly the Trigger Master designs are based on.
AWS has a FET too. And those are just the companies off the top of my head there are many more.
As for what they do, AFAIK from what I remember from my circuit design days it's basically a transistor with a gate and amplifier (in it's most simplest incarnation). In the airsoft world I believe programmable computer units were added on that do things like single shot, single with delay, 3 round burst, etc.
I believe a FET is a must when dealing with high voltage guns (ie. 9.6V+) as well as LiPo's due to the FET reducing "arcing" or something and saving the trigger assy from getting lots of carbon deposits and not working anymore.
EDIT: I just realized I assumed people knew what "assy" meant. Just in case it means "assembly" (as in "assy and details").