This is likely the down indicator used to keep a visual track of which down it is (the down marker can be wrong). By shifting the band from finger to finger after each down, the officials can immediately tell what down it is from the position of the bands.
Some umpires have another purpose for the band. They use it to keep track of where on the field (horizontally) the play started. So for a return to the previous spot (say for a penalty), there is a record of whether the play started at the left hashmark, the right hashmark or some general area of the middle of the field.
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If you are talking about the side judge, he uses that to keep track of downs, much like the Umpire having the true "game clock". *All of the officials on a football game usually wear one of the down-indicators, although some wear two (one for downs, one to keep track of where the football was positioned - on the left, right, or between the hashmarks). Also, it is the line judge who usually holds the "official" time.