Closed chain exercises are where the hands or feet are on an unmovable object, such as the floor. Examples would be squats and pushups. An open kinetic chain exercise generally has the body stationary while moving a movable object. Examples would be bench press and leg press. Generally closed kinetic chain exercises are regarded as being better for athletic gains.
So what are exercises in gymnastics rings? At first thought I'm tempted to say closed chain, but rings aren't exactly an immovable object. Yet I don't think they would fall under the category of open chain either. It's almost like they carry the benefits of closed chain exercises (like a pushup) with the added benefit of some of the stability you'd get from an open chain exercise (like a dumbbell bench press)