Execution:

The ring muscle up starts with the athlete hanging from the rings with arms fully extended. This can either be in a false grip (where the flesh of the palm and edge of the wrist are laid over the ring, leaving the thumb on the starting side of the ring) or in a neutral grip (hands simply holding onto the rings).
The false grip shortens the forearms, which improves strength, and makes the turn over from beneath the rings to on top of the rings far easier. The false grip is difficult because it is uncomfortable and many beginners find it to feel sufficiently odd enough that they do not believe they are doing it right and do not pursue it.
However according to Greg Glassman “no false grip, no muscle up” so if you want a muscle up it is time to get comfortable with being uncomfortable.

Once the athlete has mastered the false grip, the muscle up can either be performed strict; by pulling the hips to the rings until the point where the athlete’s shoulders are above the rings, then pulling the rings underneath the armpits. This is the receive position, here the athlete should be in the bottom of their ring dip position, from there the athlete drives themselves out of the dip until their arms are fully locked out.
The athlete can also kip the muscle up; using the hollow and arch kip to propel themselves up and over the rings before driving out to full lock out.

Purpose:

According to Greg Glassman the muscle up is an astonishingly difficult movement to perform but it is unrivalled in building upper body strength. In its very simplest form the muscle up simply gets you from under things to on them.

Programming:

Muscle ups are frequently used in high skill wods, as well as used in strength and skill work. As this is a very difficult skill to master there will often be skill sessions based around drills to help attain or improve the muscle up.
Including ring dip progressions, kipping progressions and banded skill work.

Variations:

The muscle up can also be performed on the bar as bar muscle ups.