The mobster tower is one of the four major battles you can take part in, the others being the gangster HQ, the sewers and the lab. Once you complete the mobster tower you get 350 money, 10 strength and 5 charm. You dont get any loot for doing this apart from stat ups so its near pointless, apart for being quite fun.
The mobster tower is located next to the cozy condo on the second island.
You need the key for the mobster tower. To get the key you need to fight a group of mobsters if you high enough strength level, make sure you have either the combat shotgun or secret weapon or something better as the mobster tower is a difficult battle.