School is a game mode where you can level up your bears. On any given day, only one affinity's bears can attend school. As you can see in the picture, the top bar tells you which affinity's turn it is today, and the button below it tells you which affinity gets its turn tomorrow. In the picture, school is in session for Brave bears today, and tomorrow Cautious bears will get their turn.
You can also click the button to see an extended schedule like this:
Bears may fall asleep after studying (leveling up) in school. If you want to have a bear study more (level up again), you can spend a Coffee to do that. Coffee levels up the bear automatically; you don't have to play a match after using it.