What exercise is this?

Well if you pull up the barbell with both hands, it's called a bent-over barbell row. It works the muscles of the back. The more your upper body is lowered, the more the muscles in your lower back are exercised. The more your upper body is raised, the more it affects the muscles in your upper back. This is true concerning any type of row - the back muscles that you are pulling the weight toward are the ones that are being the most heavily stimulated. Any exercise that involves a pull with the arms is a back exercise that also stimulates the biceps. Some people do bent-over dumbbell rows one hand at a time, and some people do cable rows one hand at a time. But I've never seen anyone lift a barbell one side at a time on the row or any other lift, with the exception of the long bar row, in which the end of the bar that stays on the floor is not loaded with weights. This strikes me as rather odd. Perhaps that particular gym doesn't have any dumbbells that are big enough for this guy!? Even then I don't know why he wouldn't just do a regular bent-over barbell row. Sounds pretty weird to me. Unless you are talking about a smith machine. You see, a smith machine and a barbell are two very different things. Were it a smith machine, one hand at a time would be perfectly normal - resting one side while you work the other, therefore burning more calories and possibly decreasing joint pain. For example, some people do one-handed upward rows on the smith machine in order to avoid pain in the wrists.