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")
In 2382, the USS Cerritos encountered a quantum fissure and was brought to a parallel universe where the crew met an alternate version of themselves. As the two crews mingled, Lieutenant Shaxs was arm wrestling his counterpart, and complimented his other self's technique, prompting the alternate Shaxs to reveal that he learned it from strangling Cardassians. The prime Shaxs gasped in excitement and stated that he acquired his wrestling form under the same circumstances. (LD: "Dos Cerritos")