Your pokemons will stop leveling once it hits the cap, depends on how many badges you have on that region. Any excessed EXP isn't wasted; it's stored and will be used once you unlock the next level cap by getting the next badge.

If you have a pokemon that's over the regional level cap (bought or brought from another region), it will directly faint as you send it to the battle. It can still use its overworld moves, nevertheless.

Kanto : 20/26/32/37/46/47/50/55/62/100
Johto : 20/24/29/32/37/39/41/46/48/55(post Ho-oh battle)/100
Hoenn : 20/24/28/33/35/38/44/48/58/100
Unova : 20/24/27/31/35/38/43/46/56/100
Sinnoh : 20/27/29/34/37/43/46/52/60/100

You can see your current cap on your trainer card. "C" button is the default key if playing on PC.