Ring push-ups train shoulder and trunk stability, and promote proper elbow tracking. Best of all, this is the most user friendly way to perform a push-up.
Have shoulder pain? The rings will allow your wrists to orient and your arms to move in a comfortable manner. Too weak to do push-ups from the ground? Bringing your body to a more upright position will decrease the intensity enough that you'll be able to do the exercise.
Push-ups too easy for you? Try doing ring push-ups with your feet elevated on a box, with a weighted vest or chains, with the straps wider, or by turning the rings out at the top of the push-up (fully externally rotating the humerus).