Arm wrestling was a physical game played on Earth in which opponents locked arms and struggled in an attempt to force the other person's hand to the table or other surface being used.
In the Fair Haven holoprogram, this game was often played at Sullivan's Public House. Harry Kim once defeated an opponent, Liam, who seemed to be much more powerful than him. (VOY: "Fair Haven", "Spirit Folk")
Klingons raised the stakes by having a d'k tahg on either side of the arms. Thus, whoever lost got their hand impaled. (TNG: "Redemption II")