Microtones are any notes that fall in between the current pitches of fixed pitch instruments such as the piano. These instruments are based on the current musical practice of "12 tone Equal Temperament". While this is a very broad way of looking at it, this is what I believe, the way most people think of microtones.
In Equal Temperament the musical octave is split into 12 equal parts or musical steps. This is a fairly recent phenomenon having come into practice over the last few hundred years. A musical temperament is a tuning system that compromises or "tempers" the pitches from a "pure tuning system", referred to as Just Intonation. Just intonation is the tuning of musical intervals as whole number ratios (such as 3:2 or 4:3) of frequencies. Any interval tuned in this way is called a just interval. Just intervals (and chords created by combining them) consist of members of a single harmonic series of a (lower) implied fundamental.
The problem with Just Intonation is that the pitches generated by the ratios of a given pitch don't always match a pitch generated by another pitches whole number ratio's which makes things more complex. The following tutorials will go into much greater detail on the explanation of this and hopefully lead to a general understanding but a quick example would be looking at the musical pitch of A. If we generated that given pitch as the 6th scale degree from the C Major Scale (the major 6th of C) we would get a slighty different note than the A that would be generated a perfect 5th above the Major 2nd of C (D).
The modern guitarist currently can find a wide range of guitars to choose from that offer "microtones". By looking at the theoretical definitions of different pitches, temperaments, and tuning systems I hope to help guitarists looking to take up this fascinating world some guidance on how to move forward.