From MTG Wiki:

A spell or ability may instruct a creature to fight another creature or it may instruct two creatures to fight each other. Each of those creatures deals damage equal to its power to the other creature.

The damage is dealt simultaneously, and it is dealt by the creatures themselves, so abilities like deathtouch, lifelink, and infect will work as normal.

Damage dealt while fighting is not combat damage, so abilities like first strike or Double strike won't have any effect.

Fighting itself does not cause the creatures to become tapped.