Mahón, the capital, is probably the most lively place on the island. In the area called Anden Poniente, close to the port, you will find a concentration of bars and taverns.
The area Muelle de Levante offers many nice summer terraces, also you may eat very good "tapas" here.

Villacarlos has nice summer terraces and restaurants as well, specially around Cala Fonts.

Alayor and Punta Prima, with fashionable discotheques, are the places for you if you want to dance.