You start off with 20 character slots. It can go over 20 characters especially when you perform a Multi-Summon. However, you won't be able tackle any quest until you bring down the number of characters to the limit. There are two ways to increase your character limit:
1. Wish it from Shenron after completing the seven Dragon Balls
2. Use a Dragon Stone in the in-game shop to open 5 character slots.