Whenever you play two notes at the same time, the difference in pitches is called an
interval. Because of how pitch perception works, generally we care about the ratio of frequencies, rather than the difference.
Some intervals sound better than others. That's subjective, of course, but it's generally held that the best-sounding frequency ratios are the ones that match up with simple integer ratios - such as 2:1 (i.e., an octave), or 3:2 (the so-called "perfect fifth"), or 4:3 (the so-called "perfect fourth" - which you may notice is also the same as going up an octave and then down a perfect fifth, since 2/1 * 2/3 = 4/3). More complex combinations are usually held to sound okay, even if the numbers get big. For example, going up two perfect fourths and then down a perfect fifth gives you a ratio of 32:27, which is sort of close to the simpler 6:5.
But none of these ideal ratios is actually produced by instruments tuned in the normal modern way (12TET). The modern tuning is designed so that the spacings are the same no matter what key you're playing in; the ratios between adjacent notes are identical. It turns out that no matter how many notes you divide up the octave into, you can't get just-intonation results if you divide it up equally. But with 12 notes (a convention with centuries if not millennia of tradition behind it), you can get very close for some notes (2^(7/12) is about 1.4983), and acceptably close for most others, and it produces a sound that everyone (at least in contemporary Western cultures) is very familiar with.
"Microtones", generally, just means "notes chosen so that the intervals between notes are smaller than in 12TET". That is to say, more than 12 notes are used per octave. (Of course, as the number of notes increases, it becomes harder to design instruments or to write or perform music.)
There are any number of reasons you might want to do this, and depending on your reasons, you might choose to space the notes out in different ways. Equal spacing with a greater number of notes produces interesting results - for example, with 19 equally spaced notes, you get an extremely accurate approximation for the 6:5 interval. Some avant-garde composers, on the other hand, choose dissonance instead; with 14 or 15 notes, the closest approximations to the perfect fifth are quite far off.
However, if we have fixed the key that we want to play in (easy enough in the case of electronic music), then instead of equally spacing the notes, we could simply add notes that are intended to make it easier to get good approximations of the just-intonation intervals in certain keys.
One example of this is in the classical Indian system, each 12TET note, aside from the root and the perfect fifth, is essentially split into two slightly different variations.