Presence of certain fins and the morphologies of each are determined by the lifestyle of each fish. I use the largemouth bass to present fin structure because it has most of fin structures of fishes.
(Picture of a largemouth bass (Micropterus salmoides) from here)
Caudal fin - used for forward propulsion of the fish.
Soft dorsal fin - may be used for forward propulsion and as a rutter to change direction.
Spiny dorsal fin - used as a keel to maintain fish balance. Can be lowered to increase streamlining of fish (increase swimming speed)
Pectoral fin - used for braking and turning while swimming and may be used for forward propulsion.
Pelvic fin - prevents fish from floating upward when fish brakes (with pectoral fins)
Anal fin - acts as a rudder for turning, may be used for forward propulsion.
(Tail of bigeye tuna (Thunnus obesus). Picture from here)
Finlets - increase speed of swimming of fish by reducing drag.
(Picture of rainbow trout (Oncorhynchus mykiss) from here)
Adipose fin - function unknown. Current though is that it does nothing, but new studies suggest that it may have sensory functions.