Introductory music theory heavily empasizes analysis and writing of music with four-part harmony, putatively the basis for music at the beginning of the common practice. Why did four-part voice leading, rather than, say, voice-leading in three, five or n parts, become the basis for much of baroque music and thereby become so heavily emphasized in music theory coursework?
This is a big question with no simple answer. One factor, though, I suspect, was the increasing trend, starting around the sixteenth/seventeenth century, for every chord to be a triad, with at least a root, a third, and a fifth, and stricter rules against parallel fifths and octaves. If you ever try composing in parts to these standards, you will quickly find that four voices is the easiest: with three, it's hard to get triads all the time without voices jumping around, and with five or more, it's hard to avoid parallels.
Modern "introductory music theory" is based on so-called "common practice" (i.e. western "classical" music between about 1750 and 1900) where the notion of "harmony" was generally more important than "counterpoint". Since the chord structure was based on stacking thirds, and 7th-chords were common, four parts are needed to produce the full range of chords that were regularly used.
There was still some consideration of contrapuntal issues (for example avoidance of consecutive fifths and octaves), but especially in instrumental music, most of the other "rules" of counterpoint were ignored in practice.
Therefore, "four-part harmony" was the simplest way to teach the basic principles, and it was also practical to perform it both on polyphonic instruments (e.g. keyboards) and on small ensembles of monophonic instruments (string quartet, wind and brass ensembles, etc) and singers.
Writing "harmony" with fewer than four parts introduces the complications of incomplete chords and "implied notes." When writing in a minor key, this can lead add a lot of problems in trying to formulate "rules" to decide whether the music is really in the minor key, or the relative major. Of course in "real music" the ambiguity may be intentional, but it is an unnecessary complication when beginning to "learn harmony".
In music for larger ensembles which is apparently in more than four parts, the additional parts were often doublings at the octave, and therefore did not require much more theoretical consideration.
In contrast to this, when the main emphasis was on counterpoint and writing for voices, rather than "harmony", fewer parts were always easier to write for. Hence teaching counterpoint traditionally began with just two parts, and generally ignored the "implied harmony" except for a few rules of thumb - for example, that the final notes should imply a perfect/authentic cadence, even though most of the notes of the cadence chords were be missing.
Relatively little counterpoint for instruments was really written in four (or more) independent parts - for example, many "fugues in four parts" by J S Bach contain long passages of music that are only in three parts, not four.
It's heavily related to the 4 major voice types bass, tenor, alto, and soprano. As melodies became more advanced and interwoven in each other, the utilization of different parts of the voice in different registers became important to distinguish between the parts.
THE RISE OF POLYPHONY
The idea of high and low pitched voices arose with the coming of polyphony in the ninth century. As polyphony developed in complexity, better educated singers were required, and one of the training devices created (by Guido D'Arezzo, 11th century) was the Guidonian Hand, the basis of a sightreading technique, SOLMIZATION, still used today. By the eleventh century, portamenti were being used on certain consonants in chant performance and the singing of descants had begun. These were elaborations performed against a cantus firmus, the protracted notes of a plainchant melody. Those who sustained the prolonged notes were called 'holders' or tenors, while those who sang the descant part 'against' them were called contratenors. The contratenors often sang the 'high' part, eventually called the altus , and, later, those who sang a part intertwining with the altos were named--predictably--contraltos. Eventually these parts were surrounded by two outer contrapuntal voices, appropriately named sopranus (above) and bassus (below).
It should be noted that a very powerful learning tool for counterpoint and voice leading is Fux's which only involves 2 voices and is a kind of precursor to the harmony used in the common practice period. With 4 distinct voices, you can cover most of the typical harmonies encountered in the common practice era and some of the more advanced ones like Neapolitan Chords and Augmented 6th chords.