Gymnastic ring exercises can be quite beneficial to people other than gymnasts. They can help improve both strength and flexibility, while being versatile enough to accommodate anyone and allowing you to move from one position to another in no time at all. For the vast majority of people, the gym rings would not be used as they would with gymnasts, as this would require learning a completely different modality. That said, they can be used in exercises that you might normally perform, but add in a novel stimulus.
Gym Ring Exercise #1: Ring Dips
The main muscles that are trained in the ring dip are the shoulders, chest, and triceps, much like a conventional dip on bars. That said, the training effect is quite different in that each repetition will require a much greater degree of upper body stability and, subsequently, strength.
Gym Ring Exercise #2: Ring Press Ups
Much like the ring dips, the ring press up targets the shoulders, chest, and triceps. That said, due to the instability of the rings, a greater degree of abdominal musculature is trained and subsequently strengthened.
Gym Ring Exercise #3: Ring Roll Outs
One of the better abdominal exercises in your repertoire would be the abdominal wheel roll outs that targets virtually the entire core musculature. Performing the same exercise, but with gym rings, trains the same muscles but with the addition of an incredible degree of instability. This serves as a novel and potent training stimulus for a strong core.
Gym Ring Exercise #4: Ring Muscle Up
The ring muscle up is a spectacular upper body exercise that is notoriously difficult to master, fundamentally combing a pull-up and a dip into one fluid movement that requires the use and coordination of virtually all the muscles of the upper body.
Gym Ring Exercise #5: Iron Cross
The iron cross is a static-strength exercise and a remarkable display of strength and control. It is a rather difficult exercise and takes a degree of dedication to learn but is not beyond the reach of the recreational trainee. If done correctly it targets virtually all the major muscle groups to some degree or another.
Bibliography
1. "23 Bodyweight Ring and Strap Exercises." The Art of Manliness RSS. Web. 9 Dec. 2013. <http://www.artofmanliness.com/2010/08/05/23-bodyweight-ring-and-strap-exercises/>.
2. "Ring training exercises." Fitstream. Web. 9 Dec. 2013. <http://www.fitstream.com/exercises/ring-training-exercises/>.