One under par for a hole is one shot less than par, say you make a 3 on a par 4, this is a birdie. One under for a round is where you had a round which was one shot less than par, say you shot 71 on a par 72.
For holes played, or for that round, or for the tournament, the player has taken one less shot to play those holes than par.
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Par is the number of strokes that it should take a very good golfer to finish a hole. If a golfer finishes the hole in 2 under par, it means he/she took two fewer strokes than par.