Completing a check or a hit on an opponent to make sure that he is unable to contribute to the play. Finishing a check on an opponent implies hitting him (usually into the boards) and not just covering or defending him, even if there might be physical contact in those cases.
Finishing the check is something a player does when defending and when he does not have possession of the puck. Therefore, it is a more common manoeuvre for defencemen than for forwards. Coaches will usually instruct defencemen to finish their checks on attacking players to eliminate them from the play and possibly cause a turnover or retrieve puck possession. Offensive players can also be instructed to finish their checks, and they do so when on the forecheck by hitting opponents who are trying to trying to clear the puck from their zone. They can also finish their checks when defending in their own zone, just like a defenceman would do.