How do level caps work in this game?

                
Your mon stops leveling once it hits the cap, set by your regional amount of badges. Any excess EXP isn't wasted, it's stored — the mon will use it to level up once you gain another badge (or become the regional Champion for the final boost) and fight something after that.
If you have a mon that's over the regional level cap (bought or brought from another region), it'll faint itself. It can still use its overworld moves.
Kanto: 20/26/32/37/46/47/50/55/62/100
Hoenn: 20/24/28/33/35/38/44/47/58/100
Unova: 20/24/27/31/35/38/42/45/56/100
Sinnoh: 20/26/29/34/37/42/47/50/60/100