In dialogue, you can sometimes choose to brawl someone. A brawl is a hand to hand fistfight between you and another character. Neither of you can die during this fight. However, if you use a weapon or cast a spell, the fight will turn into a live combat and you may be guilty of  a crime.

Brawling is not a skill and does not benefit from any of the skills. Furthermore, Brawling does not cause any skill to increase.

You will usually brawl as part of a quest objective or a plot device. For example, in the Thieves guild Quest Taking care of business, you will have to brawl with a number of people in order to complete the objectives of that quest.

Khajiit is the best race for brawling, since they have a natural, unarmed, claw attack. The Heavy Armor perk Fists of Steel lets your heavy armor gauntlets increase the damage of your unarmed attacks.

If you are losing a brawl, do not be tempted to draw your weapon or cast a spell. You cannot die in a brawl, however drawing a weapon or using magic will turn the brawl into a real fight, and most likely cause you to fail the objective as well as getting arrested.