A crab was a crustacean native to Earth.
When Spock tried to match the sound of the Whale Probe to the sound of an Earth animal aboard the Klingon Bird-of-Prey HMS Bounty, a rock crab (Cancer productus) and a Vulpes velox were two of the animals shown. (Star Trek IV: The Voyage Home)
Crabs were Ferengi delicacies. Grand Nagus Zek's son Krax was fond of crabs. (DS9: "The Nagus")
Keiko O'Brien often made crab rolls. (DS9: "Tacking Into the Wind")
In the original script for "The Cage", Talos IV was named Sirius IV and was inhabited by crab-like aliens who kidnap Christopher Pike.