Male Sand Fiddler Crab
Many of these were displaying on the edge of a salt marsh at the nature boardwalk at Huntington Beach State Park, South Carolina. This is a good site for birding and nature watching, with an excellent interpretive center--see Bill Hilton's article.
The crab's display is a hilarious "wave" of the large claw into the air, and the males seemed to do it in bouts--one would start, then they would all respond. Most of the waving appeared to be directed at other males, but once I saw a female approach a waving male. He waved furiously, then she approached closely and stood next to him as he placed the large claw gently on her side. It was hard not to anthropomorphize that scene. (Unfortunately, my photos of the happy couple were not great.) I had to stalk close and stand for about 15 minutes to get these photos--the crabs were skittish, and dove into their burrows if I moved quickly.
They appear to be the Brackish Water Fiddler Crab (also called Red-jointed Fiddler), Uca minax, based on images I can find on the Internet and Gosner, Field Guide to the Atlantic Seashore, which keys the three widespread eastern species (I'm going to double-check this--could also be the Sand Fiddler, U. pugilator). Links:
(My latest thinking is that this is a Sand fiddler, Uca pugilator.)