In the common 12-edo tuning, the octave is divided in 12 equally-spaced steps (hence equal divisions of the octave). What's meant my “equal” is that the frequency ratio between subsequent notes is always the same, or equivalently the logarithm of the frequencies of neighbouring notes always have the same difference. The other way around, it means that the frequency of the i-th note can be computed as an exponential, specifically
fi = f0 · 2i⁄12 = f0 · (12√2)i.
Tabulated, these frequencies are, starting from A440,
440.0 Hz, 466.2 Hz, 493.9 Hz, 523.3 Hz, 554.4 Hz, 587.3 Hz, 622.3 Hz, 659.3 Hz, 740.0 Hz, 784.0 Hz, 830.6 Hz, 880 Hz
The main reason 12-edo is so common is that it is, without too many possible notes, a reasonably good approximation to 5-limit just intonation. For example, seven steps come out as
f7⁄f0 = 27⁄12 ≈ 1.4983
which is very close to 3⁄2 = 1.5. Therefore, a 12-edo perfect fifth sounds almost exactly like a JI fifth. It is slightly narrower, but almost imperceptibly.
Not quite as good is the approximation of thirds:
f4⁄f0 = 24⁄12 ≈ 1.2599
That one is audibly wider than the JI major third 5⁄4 = 1.25, but it still good enough to pass off as an approximation in many contexts.
12-edo is by no means unique in offering approximations to those JI intervals. With more steps, you can in particularly get better thirds, though the fifths typically get a little worse. Specifically, the ratios for fifth and major third in the best 5-limit tunings are
Because Western tonal music is essentially 5-limit, all of these tuning systems can be used for rendering most music, though there are various quirks to be aware of. 19-edo and 31-edo are quite easy in the sense that like 12-edo they're meantone temperaments, meaning a major third (i.e. the approximation of 5⁄4) has the same size as two whole steps (i.e. the approximation of 9⁄8). Obviously, this is not the case in just intonation, and not in 22-edo and 34-edo either. Particularly in 22-edo, the ditone comes out as notably wider than the major third, which may result in unexpected asymmetries in melodies, which can be a difficulty but also an opportunity for the composer.
In meantone tunings it's generally quite straightforward to render music in standard notation, because the intervals can be read off and each interval has a clear correspondence. Typically, e.g. E♭ be different from D♯. In non-meantone tunings, even two E♭ notes may be different, depending on context (basically, whether they're approached by a ajor third or by two whole steps). Via modulations (look up comma pump) that can even happen even in meantone as well.
Unlike 12-edo, the higher EDOs also approximate JI ratios with prime factors higher than 5, for example 31-edo includes a Barbershop seventh 1.7489, very close to 7⁄4 = 1.75. This may or may not be exploited when composing music for those tunings.