This article is from the Dolphin
FAQ, by Jaap van der Toorn
Dolphins communicate mainly by means of sounds. These sounds include whistles, but also so-called pulsed sounds, which are often described as squawks, barks, rasps, etc. But they also use breaching (jumping and falling back into the water with a loud splash) and pectoral fin (or flipper) and tail (or fluke) slaps (hitting the flipper or fluke on the water surface). Body posturing and jaw popping also have a role in communication. This list is not exhaustive.
As for language, we do not know if they have one. Several studies have demonstrated that dolphins can understand a structured language like ours. This same has been demonstrated for a number of other animals species as well (gorilla, bonobo, California sea lion, parrot). Some studies also indicate that dolphin vocalizations are complex enough to support some form of language. However, to date it has not been demonstrated yet that they indeed use a language for communication among themselves.
R.P. Balda, I.M. Pepperberg & A.C. Kamil (eds.) (1998) Animal cognition in nature - The convergence of psychology and biology in laboratory and field. Academic Press, San Diego, London
R.J. Schusterman, J.A. Thomas & F.G. Wood (eds.) (1986) Dolphin cognition and behavior: a comparative approach, Lawrence Erlbaum Associates, New Jersey