The family is called Delphinidae.
Within this family, there are about 40 different species of dolphins. Some common examples include:
* Common dolphin (Delphinus delphis)
* Bottlenose dolphin (Tursiops truncatus)
* Atlantic spotted dolphin (Stenella frontalis)
* Killer whale (Orcinus orca) (although often called a whale, it's actually a dolphin!)
So, to specify the genus and species, you would need to name a specific type of dolphin, like "the genus and species of the common dolphin is *Delphinus delphis*."