In My Little Pony, "moons" are often referenced as a length of time. While most of these uses are completely useless in determining how long a moon is, one specific use of "moons" can easily tell us exactly how long a moon is.
In the season 4 premiere, Celestia states that the Summer Sun Celebration will be the first in which it isn't a celebration of Nightmare Moon's defeat, but of her sister Luna's return. Since the series started on this same celebration, it's likely that this is exactly a year later. With this knowledge, we can safely say that the entirety of Seasons 1 through 3 (and Equestria Girls, sort of) take place within a span of 365 days (assuming Equestrian years are the same as our own, which they appear to be).
In the episode Apple Family Reunion, Granny Smith says the reunion happens every 100 moons. However, the last time we saw this reunion was in Friendship is Magic, Part 1, almost a year prior. This means one simple thing: 100 moons is a bit less than 365 days.
Assuming that the name "moons" isn't entirely arbitrary and the length of a moon is related to the moon itself, then the two most likely values for one moon are:
- One day (unlikely), or
- One phase of the moon (likely).
One phase of the moon takes around 3.4 to 3.6 days (I've found competing numbers but I personally calculated it to be 3.42), meaning 100 moons is about 340-360 days, or just short of 365. I haven't found any evidence to prove that a moon is any other length of time (except in the comics, which contradict canon on multiple occasions anyways).