Male crickets fight over possession of a territory or a female. Fights usually begin with antennal fencing (lashing of one another’s antennae). Males may exhibit a threat posture (raising themselves on the forelegs). They commonly kick one another, and lunge at their opponents, often with their mandibles flared. Stridulation, a form of aggressive communication (performed by rubbing the wings together), is usually a component of fights. The most aggressive behavior in a cricket fight is grappling, in which males engage their mandibles and attempt to flip their opponents on their backs. At the end of the fight, the loser usually flees and the winner performs a rivalry (victory) song. Winner/loser effects have been observed, in which winners tend to win subsequent fights, whereas losers tend to lose subsequent fights for up to several days.
You'll find more videos of aggressive interactions on the Cricket Videos page but here are videos of Acheta domesticus fighting:
and Gryllus integer fighting: