mimeTeX is a program written by John Forkosh, thankfully made available under the GPL license. You can read all about it and download the original source code from http://www.forkosh.com/mimetex.html
mimeTeX is used in Moodle as the main engine that converts TeX code into graphic images. It's used by the tex and algebra filters.
To work, mimeTeX needs to be compiled to run as a binary. Because many people don't have the facilities to compile this themselves, this directory contains pre-compiled binaries for various platforms.
To install them in Moodle, simply download the 'mimetex' that most closely corresponds to your platform, and drop it into your filter/tex directory.