The term "Atlantican" in The Little Mermaid trilogy means that one is from the kingdom of Atlantica, the undersea kingdom of Ariel's father, King Triton, and her late mother Queen Athena. Atlantica is the home of all the merpeople mentioned in The Little Mermaid trilogy.
Before The Little Mermaid (film), it is revealed that Atlantica was an overwater kingdom called Atlantis and filled with merpeople and centaur, until Princess Ariadne fell in love with a centaur named Jason. When the shark Ariadne was betrothed to found out, he erupted the volcano, making the island break and forcing the centaurs to move to Mount Olympus. This was all revealed in Of Myths and Mermaids