To play a module file, you'll need a tracker or player program if you don't already have one.
VLC Media Player
I guess VLC media player should be just right for playing mod files as well as MP3. In fact it plays just about anything. It is available for Mac OS X, several Linux distros and Windows.
Linux users are happy, XMMS plays them all. :-) I've noticed this by accident while evaluating a linux distro. XMMS does handle a lot of file formats, additionally there are a growing number of plugins available for playing OGG, SID, Amiga and Atari sound files.
CocoModX is a freeware audio module player for Mac OS X.
It supports almost any module type including zipped ones, and is under steady development. Mod software for Macintosh used to be very hard to find, so thumbs up for this player.
An alternative might be the VLC media player, which I've seen playing several MOD files and doing a quite good job at it.
XMPlay and BASS
That XMPlay thing has been my current favorite player for Windows. It plays many module formats, MP3, OGG, WMA, and I use it for streaming internet radio too. It doesnt clog your system like some other players. And it's free and without ads and gimmicks. Another feature: Linux users using XMMS can use XMPlay's playback engine BASS as a plugin. The BASS Developers Kit contains all sound supporting resources you need when programming in C++, Delphi or Visual Basic.
Coders should also take a look at FMOD, a cross-platform sound engine which is free for non-commercial use.
MODPlug Player and MODPlug Tracker
Among the Windows-based module players the MODPlug Player by Olivier Lapicque is quite popular with a long tradition. Most of the time, it reads the effects and instrument data correctly, which is a big plus. The MODPlug programs have their own base on the web at www.modplug.com.
MODPlug Player is one of the best known MOD players, as it reads a great variety of MOD formats, while the ModPlug Tracker is a nice proggy too. By the way, the ModPlug engine is open source.
ImpulseTracker and the like
Now, a toast to good old ImpulseTracker by Jeffrey Lim.
This software can do more neat things than it seems, and the only reason why I don't provide a link is that its site is gone. I had it running on a DOS box since 1995 and just couldn't abandon it.
It runs under Windows 2000 and XP quite smoothly if VDMS is installed.
Later, chisel/Storlek built Schism Tracker which brought the look and feel of ImpulseTracker to life in an open source, cross-platform environment. I am very thankful for that.
Both Schism Tracker and Impulse Tracker owe their existence to Scream Tracker (by Future Crew) and to some point FastTracker II (by Triton)
Psycle is a free open-source tracker. One major advantage of this tool is that it supports VST Plugins and VST Instruments as well, a great feature which is not yet quite common among free trackers.
Psycle was created by spanish scene fellow Arguru, who passed away in 2007.
What was formerly known as NoiseTrekker eventually became ReNoise. This is a powerful tool with the feel of FT2, now featuring Full MIDI, VST/VSTi support and realtime DSP. It is quite easy to learn and very useable. You can try it out for free, and registering the software isn't too expensive either.
VDMsound, yay! This was mentioned earlier. VDMSound (or VDMS) is a tool that was made to help run old MSDOS games in newer Windows environments, above all if there are problems with sound output. That said, it is also possible to run good ol' ImpulseTracker and FastTracker II under Win2k with VDMsound.