24-EDO tuning is 12-EDO tuning with all of the notes split in half.
This tuning is common in eastern music. It's close relation to 12-EDO makes it easy to notate. By adding a half flat "d," and a half sharp "t," the notes can be listed as:
A: 0 cents (unison)
At: 50 cents (diminished 2)
A# / Bb: 100 cents (minor 2)
Bd: 150 cents (neutral 2)
B: 200 cents (major 2)
B# / Cb: 250 cents (augmented 2 or diminished 3)
C: 300 cents (minor 3)
Ct: 350 cents (neutral 3)
C# / Db: 400 cents (major 3)
Dd: 450 cents (diminished 4 or augmented 3)
D: 500 cents (perfect 4)
Dt: 550 cents (harmonic 11)
D# / Eb: 600 cents (augmented 4 or diminished 5)
Ed: 650 cents
E: 700 cents (perfect 5)
E# / Fb: 750 cents (diminished 6)
F: 800 cents (augmented 5 or minor 6)
Ft: 850 cents (neutral 6)
F# / Gb: 900 cents (major 6)
Gd: 950 cents (diminished 7 or harmonic 7)
G: 1000 cents (minor 7)
Gt: 1050 cents (neutral 7)
G# / Ab: 1100 cents (major 7)
Ad: 1150 cents (augmented 7)
New intervals are allowed, such as the neutral intervals, which greatly expand the musical palette, but these new intervals are unfamiliar to audiences exposed only to western music. Studies have been done, however, to show that babies are able to vocalize neutral intervals like the neutral second and neutral third, before they are able to vocalize all of the intervals of the major scale.
An example of a scale that makes use of these tones is the Rast scale:
1 2 d3 4 5 6 d7
In the key of C: C D Ed F G A Bd C
In the key of A: A B Ct D E F# Gt A
On paper, the scale looks like a cross between the major scale and the harmonic minor scale, but the sound is quite different from either.
24-EDO is a great place to start with microtonal music, as there are plenty of examples of the tuning in use in eastern cultures, and as all of the tones from standard western tuning are available in the same tuning.
Older Persian and Babylonian musicians used a tuning that was similar to 17-EDO, but tuned with just ratios rather than pure equal octave divisions. The tuning has been almost completely replaced with 24-EDO in modern non-experimental practice, but it should be noted that 17-EDO is still an interesting tuining, particularly for those with little experience with microtonal music.