Microtonal

Music using microtones; musical intervals significantly smaller than a half step. (E.A. quarter tones or smaller)

/\ that is the most precise definition, but more generally it can refer to any tuning system other than 12 tone equal temperament, usually not including those that sound like it. (well temperaments)

Arabic music doesn't use quarter tones, contrary to popular belief. Arabic music uses neutral intervals. (half way in between major and minor) For example, the neutral second is approximately 3/4 tones.

Dude! Arabic music is sick, with all those quarter tones and stuff.

Arabic music doesn't use quarter tones, the smallest interval used is a half step; Arabic music isn't really even microtonal. Go listen to some Greek enharmonic scales or Charles Ives!

microtonality

The use of musical pitches outside of 12 tone equal temperament (these are called microtones).
While microtones technically refer to intervals smaller than a half step (a standard note in 12TET), they can also commonly refer to intervals greater than 100 cents, such as 150.

A better word for microtonality is xenharmony (microtonal -> xenharmonic).

When talking about microtonality, changing the base tuning (A = 440hz) is not commonly considered microtonal, as the intervals between all of the notes remains the same.
Also, the use of just intonation is not always considered microtonal either.

Microtonality is most commonly utilized by changing the amount of notes per octave to be greater or less than 12.
It is also common to use a "subset" of a tuning with many notes, like 53TET, because it is hard for humans to hear small differences in pitch.

12TET = The most common musical tuning on the planet. Commonly called "western" tuning. It utilizes only 12 notes per octave. (that's all the notes on a piano.)
TET = Tone equal temperament. It refers to how many equal notes are per octave of a tuning.
EDO = Equal divisions of the octave. The exact same thing as TET.
Octave = The same musical pitch, but doubled in frequency. We perceive this as the same note, just "higher" or "lower."

1: Did you see Sevish's new album yesterday? (it uses a lot of microtonality)
2: Who is Sevish?
1: Only one of the best microtonal musicians of all time! He's up their with Xotla and King Gizzard & The Lizard Wizard.
2: Wow! I should go listen to some of their music!