The Brawl Index, often abbreviated BI, of a power is the amount of damage a power does, expressed as a multiple of the base damage that the Brawl power does. It is used to compare the damage potential of different powers against each other. The Brawl power is available to every archetype and does very little damage, which is why it is used as the standard on which the damage of other powers is based.
For example, if the Brawl Index of a power is 2.5, then that power will do 2.5 times more damage to a given enemy than the Brawl power. If the Brawl power, without enhancement, would do 10 points of damage to the enemy, then a power with a Brawl Index of 2.5, without enhancement, will do 25 points of damage, assuming the enemy has no resistance to the effects of the Brawl power or the power being used. A power that has a Brawl Index of 2.5 is 25% more powerful than a power that has a Brawl Index of 2.0.
Note that the actual damage inflicted using various powers varies according to enhancements, damage buffs, enemies' damage debuffs, and enemy resistances.
The concept of the Brawl Index is entirely player-created. The developers do not reference or go by the index as such.
While Brawl Index can be used to compare the relative damage of different powers belonging to the same Archetype, it is less useful comparing between Archetypes since Brawl itself may do different amounts of base damage.
Furthermore, Brawl Index neglects the animation time needed to use a power. For example, Flurry does fairly high damage, but a different character could perform 2, 3, or even 4 other attacks in the 3.07 seconds it takes to unleash a single Flurry (not including recharge).
Due to both of these issues, Damage per Cast Second (absolute damage divided by the length of the attack animation) is now considered to be the preferred basis for power comparison.