When a player uses a creature to fight, the player exhausts the creature and chooses an opponent’s creature. Both creatures deal an amount of damage equal to their power value to the opposing creature in the fight, and both are “fighting” for the purposes of card effects.
After a creature is used to fight, if that creature survives the fight, all "Fight:" abilities on that creature then resolve. If either creature in a fight has a constant ability referencing the end of the fight (example: “after an enemy creature is destroyed fighting this creature…”), the creature must survive the fight to resolve the ability. "Fight:" abilities only trigger for the creature that was used to fight, not for the other creature in the fight.
If either creature in a fight is destroyed while resolving Assault, Hazardous, or "Before Fight:" abilities, then the fight (dealing damage based on power) is skipped. The creatures are not considered to have been in a fight for the purpose of card effects that reference “fighting” or “in a fight”, and the attacking creature’s “Fight:” abilities will not trigger. Card effects that reference “after a creature fights” or “after a creature is used” will still trigger, as the creature was used to fight (even though the fight did not resolve).