If you click on the red "battle" button at the bottom of all non-game screens it will bring you to the battle menu.
From here you can select opponents to battle. They are divided into two types: Heroes and Villains!
Heroes are available certain days of the week and you can battle against them to earn various types of ISO-8 that you can use to increase your Tsum's power
Villains are normally available for a limited amount of time (and sometimes only for limited hours during the day). The currently available Villains will change over time. When you fight against a Villain there are two main things you can get that you can't get otherwise. First, you can get Violet ISO-8 which is required for the higher levels of character power. Second, you can also get a box that contains that Villain as a playable character!
We keep track of the current battle schedule on our Battle Mode Schedule Page here!