It is very similar to Last Titan Standing, in where everybody spawns in a Titan and has to kill the enemy Titans. However, there are respawns and instead of having to eliminate all Titans to win, you must get the most titan kills. In order to win, your team must get 50 kills (Lowered to 30 kills as of Monarch's Reign).
Up until The War Games, Titan Brawl was a featured-only gamemode that rotated its way into matchmaking every few weeks. However, by popular demand, the gamemode is now a permanent staple of Titanfall 2 matchmaking, replacing the underpopulated Free for All playlist. The update also added the ability to see how much damage you have dealt to enemy Titans in the scoreboard (like LTS), as opposed to tracking assists.
Aegis Titan Brawl Edit
As of Postcards From the Frontier, Titan Brawl has it's own sub-variant, called Aegis Titan Brawl. Like the similarly-named Aegis LTS, Aegis Brawl allows players to use their unlocked Aegis Rank upgrades outside of Frontier Defense. This mode was featured-only, however.