Many musicians wanting to get into microtonal music as a broad perspective will choose to take the standard notes and subdivide them into quartertones, yielding 24 total notes, all equally spaced, which is called 24-EDO for short.
So, in 24-EDO, you have the same standard notes: A A#/Bb B C C#/Db D D#/Eb E F F#/Gb G and G#/Ab. Twelve notes with seven natural notes and five sharps and flats. Then you add the in-between notes, half-sharps and half-flats.
A
At (half sharp)
A#/Bb (enharmonic equivalents)
Bd (half flat)
B
...
Then it gets tricky. What do you do with the note halfway between B and C? From the tonal center of A, the interval is halfway between a minor third and a major second, so, it's not a neutral second or neutral third, and in our 12-EDO stricture, we already have defined augmented and diminished intervals for these. It's just kind of a clashy oddball interval.
So, 24-EDO is great for microtonal novices, because it preserves 12-EDO standard and adds onto it, but it also presents some challenges that need to be kept in mind when composing and performing. Many players are dealing with this by mixing some 24-EDO notes with the 12-EDO standard, cleaning up some of the mess and cutting down on note clutter.
I've tried 24-EDO, 12/24-hybrid, and 19-EDO, and my honest opinion is that 19-EDO is just much simpler a system than 24-EDO. You get all of the advantages of using equal temperament, you get extra intervals to play with, and you get everything in this nice package for western music theory and notation. 24-EDO has a long history in eastern music...well, sort of. Really, most eastern music using microtones does use something like 12/24-hybrid.
As far back as quartertones go with eastern music, 19 tone music in the west goes back just about as far, but it just got tucked away into the shadows of music theory and punished more severely, mainly due to the conservatism with the standardization of western music traditions.