A tracker is a type of music creation program. They're typically used on hobbyist music, because they sound extremely good considering the fact they're freeware or shareware, and can run on cheap hardware (almost any computer these days is fast enough to mix dozens of PCM tracks on software). Nothing stops people from using them for high end, though.
The trackers operate on patterns. A pattern is, simply, a fragment of note data. It's usually entered as a long list of rows (typically some power-of-two amount of rows per pattern - many trackers seem to use 64 rows by default). The rows are iterated over at constant speed. Thus, basically, trackers merely tell what instruments are played and when, but aren't specific about the lengths - no "A quarter note here", but rather "When you arrive to this row, this sound is played on this channel."
Each row has slots for number of channels. First trackers (SoundTracker, Noisetracker, Protracker etc) used 4 channels (this was just the hardware limit of early Amiga hardware), and latter trackers used 6, 8, 16, 32, and even more channels.
Each "note" on row encodes the note value (often represented as NSO where N = note, S = # mark or - depending on if the note is sharp or not, and O = octave - example: C-4 as the middle C, not to be confused with the explosive), often volume setting, and effect. There may be only an effect on the note if you only need the effect - for example, you may play a note on one row and use "slide down" effect for the next 15 rows to create a pitch bend effect.
Exact value and function of effects depends on module format, but often the effect is represented as CNN where C = command, and NN = hexadecimal numerical parameter. (Note that hex numbers are usually used for everything in trackers, except for some values like tunings...) The effects represent things like panning, pitch changes and slides, volume changes, vibratos, and even SID-style arpeggios.
Okay, now we have a pattern, on which we have note data and effects. After this, come samples. Early trackers used simple samples as instruments, but later trackers (FastTracker in DOS, at least) started using more complicated instruments - multiple samples could be assigned to single instruments for specific pitches. For example, for maximal sound quality you could sample every sound that pianos could produce and assign those to specific pitches - this sounds better than the crude pitch changes used by earlier trackers, but it understandably takes more space. =) Most tracker users seemed to use octave-grade samples. These days, some trackers are also capable of producing MIDI output.
Patterns, then, with all this instrument information, could be ordered into modules. Basically, now you have pieces of music, so it is time to arrange all that into a coherent whole. (Or so the theory goes). These modules could be then saved to disk (patterns, sounds, everything, into one file), played, or used for harassing neighbors.
Some common module formats include .mod (Amiga trackers), .s3m (ScreamTracker 3), .xm (FastTracker II) and .it (Impulse Tracker).
There are also different kinds of programs that draw their inspiration from tracking but expand the concept; Jeskola Buzz is one of such programs.
Now, don't ask me how to use these things creatively, I'm more of a sound technician rather than a musician...