The caudal fin, or tail, of a fish is the only fin to be connected to the vertebral column.  It is the primary means of locomotion for most fish.  Unlike many marine mammals with tails that use an up-and-down motion, fish generally use a side-to-side thrust of their caudal fin for propulsion.  The shape of the caudal fin can be indicative of the style of motion for a fish.  

Caudal fins come in many shapes and sizes; form often follows function

For example, very fast swimming fish like tunas have lunate caudal fins for more efficient swimming while lie-in-wait predators like grouper have a strong, wide base of the tail, the caudal peduncle, for faster acceleration. (fishionary.fisheries.org/caudal-fin)

A caudal fin graced the Summer 2012 cover of Eddies magazine (fws.gov/eddies)