Summarized from wikipedia
Module files (MODs) are a class of file formats used to represent music on a computer. They store several patterns or pages of music data in a form similar to that of a spreadsheet. These patterns contain note numbers, instrument numbers, and controller messages. The number of notes that can be played simultaneously depends on how many tracks there are per pattern.
Module files also give a list of the order in which to play the patterns. However, the biggest advantage of MOD family over standard MIDI files is that MODs include their own audio samples and should sound exactly the same from one player to another barring interpolation methods and any errors in players.
Module files are often referred to as tracker modules, and composing modules is known as tracking, simply because the first ever module creating program was Soundtracker, created by Karsten Obarski in 1987. Soundtracker was cloned many times, with programs such as NoiseTracker and ProTracker being direct descendants from the original Soundtracker code, and others such as MED/OctaMED and Oktalyzer being written from scratch. Such programs are called trackers in general.
With the advent of lossy compression and broadband Internet connections, module files are becoming less popular as more and more high-quality free music comes in the form off lossily compressed files which take acceptable time to download.
Module files comes with various extensions, according to the tracker (i.e. module editor program that created it). Some examples are: MOD, 669, MTM, STM, S3M, ULT, XM, IT, FAR, WOW, OKT, DMF, MED.
Most media players can play module files with the proper plugin(s); some media players might even support playing module files out-of-the-box, but perhaps not all module formats.