Brawl and Training are two game modes that can be found in the Brawl building on the main screen.
In Brawl, you battle other players without the fear of losing your Fame, as opposed to Hall of Fame. If you lose, you lose nothing. If you win, you receive amazing rewards that can include Event Runes, Gold, Soul Dust, Hero souls and more.
In Training, you fight against your own Squads to test their strengths and work out a strategy that suits your Heroes of choice better. This is a great help in planning your battles in other game modes.