Soft-shell blue crabs are those that have recently molted (shed their old shell in order to grow larger). Immediately after molting the new shell is temporarily paper-thin and very soft to the touch.
Many consider soft-shell crabs to be a delicacy and they are often sold at fish markets and at seafood restaurants. Soft-shells are the featured ingredient in the "spider roll" featured where sushi and sashimi are served.
If you happen to catch some of your own, check out the internet for easy to follow videos detailing how to clean, prepare, and cook soft shell blue crab.