Fight is a keyword action that covers a concept that has been around for awhile. It means that two creatures each deal damage to each other equal to their power. Its pretty straight forward, and the word ‘fight’ is a lot shorter than saying something like “these two creatures deal damage to each other equal to their power.’ Two newer examples of cards with this keyword action are Nightfall Predator and Prey Upon. Some cards with similar abilities have received errata so they now use this keyword, for example; Contested Cliffs and Magus of the Arena. There are a few rules related aspects to be aware of with this action.
The first is that if when a spell or ability resolves a creature that is no longer in play is instructed to fight, nothing will happen. A creature that is not in play cannot deal damage via fighting and it cannot receive damage. This works somewhat similarly to how exchanges work, if one of the objects is no longer on the battlefield it cannot be exchanged. It one of the creatures is not around, it cannot fight.
It is also important to note that damage dealt when two creatures fight is not combat damage. Abilities that trigger when a creature deals combat damage will not trigger.
Finally, if a creature is instructed to fight itself it will end up dealing damage to itself twice. In other words, it will deal damage equal to twice its power to itself.