This page will tell you everything about MUGEN and what it is.
What is MUGEN?
MUGEN is a 2D fighting games engine made by a company called Elecbyte in 1999. Originally starting as a SHMUP game, Elecbyte soon realized that there weren't any good commercial fighting games for the PC. Taking inspiration from a Korean Street Fighter 2 pirated game called SF2IBM, Elecbyte change the game from a SHMUP to a fighting game, so came MUGEN on the MS-DOS - which predates the very first version of Windows. You can read more about MUGEN's history in this article.
People from all around the world make characters from different games, like Mario, Goku, Ryu, even Homer Simpson, but not only do they make characters, they make stages, lifebars, and screenpacks too.
A screenpack? What's that?
A screenpack is a thing that changes the screens in your MUGEN. The screens in question are: The title screen, the options screen, the character select screen, and the vs. screen. Elecbyte added a victory screen in MUGEN 1.0, which was released in 2011, so screenpacks added that to the list.
Where can I get MUGEN?
Google has many links to MUGEN, but the MUGEN Guild and the MUGEN Free for All forums have links to WinMUGEN, MUGEN 1.0 and the currently unfinished MUGEN 1.1.
Where can i find the characters and stages, and how do i install them?
As I said before, Google has many links. MUGEN Free for All has collections of characters and stages from different series or franchises, 7zip should be able to extract the files of the characters you're looking for. There are 3 different files for the characters and stages: zip, rar, and 7Z. 7zip can extract all 3 files.
What if I want to make my own characters and stages?
There are tutorials on YouTube, the MUGEN Guild, and MUGEN Free for All, but I suggest following a guy who I consider as my teacher, Ryon Persaud. Though I only learned how to make projectiles and how to replicate Ryu's Tatsumaki and Shoryuken, there are other tutorials out there that Ryon didn't make, but he is open for request.
Are there any full games made with this engine?
Yes, I actually know some.