Turns are required to do any battling in the game. You can find your turn counter by the sword icon on the status bar. Each character gets 2 new turns at every hour change.
Each new character starts with 70 turns, except at the launch of the age. There is a staggered start for the first 20 hours of each age, so characters created in the first hour will have 30 turns, then those created in the second hour will have 32 and so on. This helps make it so no one loses turns because they can't be on when the game starts.