Why use alternative tunings?
When musicians are first introduced to different tunings, they often respond "why bother"? The following are some of the most popular reasons to explore tuning.
For something new
99.9% of music heard in the western world has one thing in common: the same 12 notes, and, thus, the same available chords, chord progressions, and scales. Alternative tunings offer an infinite number of all those things, with as many (or as few) notes as you want, in whatever relationships to each other that you want. Using only 12-tone equal temperament in spite of so many other tunings is analogous to using only major scale, in spite of other modes like minor, mixolydian, dorian, etc.
For harmonic purity (or not)
For various historical and practical reasons, the standard western tuning is arguably "out of tune" (based on our perception of consonance). For example, the major third is 14 cents sharper than pure, and the minor third is 15 cents flatter. There are also pure intervals that 12 equal divisions of the octave does not even come close to approximating, intervals most people have never heard in or out of tune before. These purer, more in tune harmonies provide a desirable sonic quality ("blending", "buzzing", "sweetness", etc), and alternative gives you access to these pure harmonies. There are also much more dissonant intervals than those that exist in standard western tuning, and alternative tunings can give one access to those as well.
For new melodies
The scales available in 12 equal divisions of the octave are relatively few. Most commonly, musicians will use one of two modes of the "diatonic scale". If you want, you can play in tunings that have structures similar to those in 12 equal divisions of the octave, but have them tuned differently so that, for instance, the difference between major and minor becomes more or less pronounced. There are many other possible scales, however, and most of them are completely unsupported by our standard 12 notes. While our diatonic scale is defined by 5 large steps and 2 small ones, there are other scales defined by any other combination of large and small step sizes. (For example, 4 large and 3 small steps defines what's referred to as a Hanson scale, and 2 large and 5 small steps defines what's referred to as a Mavila scale). Some scales are not defined by numbers of step sizes at all, and instead are derived by including the purest, most in tune harmonies. These scales tend to have very irregular melodic spacing, which gives a completely different feel. Of course, these are just two of an infinite number of possible approaches to creating scales with new melodic structures, all of which would be impossible to play in standard western tuning.
For historical purposes
It was only after the 1800s that 12 tone equal temperament became the ubiquitous tuning it is today. Before that, tuning was as much an artistic decision for a composer as any other. One could argue that music written in earlier time periods is most authentic when played in the tuning for which it was written, and that part of the music is lost when played in standard western tuning.
One example is "Well Temperament", a tuning with 12 notes that are spaced un-evenly. This tuning gives each key its own "flavor", and was the inspiration for Bach's "Well Tempered Clavier", where a piece was written in each key to show off its unique properties. A precursor to well temperament was the Meantone system, a method of tuning that, by varying the tuning of all fifths in the tuning, included tunings like 19 or 31 equal divisions of the octave. Even before that there was Pythagorean tuning, with thirds even more out of tune than those in our current standard tuning. These intervals were used as restless dissonances that resolved to the more stable, pure intervals in that tuning, like fourths, fifths, and octaves. As you go back in history, you find more and more systems of tuning that fit the music of the time period.
For cultural purposes
12 equal divisions of the octave is not standard everywhere in the world. Other cultures have developed their own tuning systems, many of which can not be approximated by 12-edo. If one wants to authentically play the music of these other cultures, then one must emulate the tuning along with the instruments and style.
Maqam, an Arabic musical style, uses so-called "neutral" intervals that can't be approximated on Western instruments because they are right in between the standard western notes. Gamelan, an indonesian ensemble and musical style, uses two separate tunings that both deviate drastically from western tuning. One is close to 5 equal divisions of the octave, the other is close to 9. Many African percussive marimba-like instruments use a tuning that is well approximated by 7 equal divisions of the octave.
To temper out commas
In some cases, quirky tuning coincidences pop up. For instance, in standard western tuning, adding 3 major thirds gets you to an octave. We've already addressed that these thirds are "out of tune", but if you want three major thirds to equal an octave, then 12-edo is the tuning for you, because with pure thirds this would not be the case. There are many many other tuning coincidences that require tunings other than 12-edo. For example, if you want your minor third divided into two equal pieces, 12-edo doesn't do it, but 16-edo (among others) will.
These coincidences are most often described in terms of "Just Intonation", the classification of intervals by the ratio between the two frequencies they're made of. The major third in 12-edo can be described as a sharp version of the interval 5/4, and the octave can be described as a pure 2/1. When we say that 3 major thirds are equal to an octave, we are also saying the following:
Which is the same as saying the following:
Small, close to 1/1 intervals like 128/125 are referred to as "commas". When one equates a comma with 1/1 (the unison), this is referred to as a "tempering out that comma". By tempering out a comma, you are equating all intervals that differ by that comma (like three major thirds and the octave in the case of 128/125.) After tempering out a comma, a sequence of intervals that normally takes you away from your tonic (either in Just Intonation or in another tuning that doesn't temper out your comma) can end on the same note you started on! This can be an inspiring compositional feature; and using it in this way is referred to as a "comma pump."