A dancer was a person who danced either professionally as a form of entertainment, or non-professionally for recreational or formal purposes.
Dr. Beverly Crusher was an award-winning dancer, having come in first in at least one tap and jazz competition at Saint Louis Academy. This earned her the embarrassing moniker "The Dancing Doctor", which she tried to live down and keep quiet. She was also capable of teaching the waltz. (TNG: "Data's Day")
In discussing a Bajoran leader who might be a good choice to meet, Crusher spoke of Jas Holza. She stated that he was a concerned leader and good spokesman for his people and that she had found him to be a terrific dancer at a reception afterwards. Ensign Ro Laren, however, was dismissive of the suggestion, describing Holza as a token Bajoran with no real influence among her people. She recommended instead a meeting with Keeve Falor, who she stated had no diplomatic experience and "won't ask you to dance." (TNG: "Ensign Ro")
In 2372, a dancer at a bar on Sobras challenged her patrons by offering up a puzzle, and the first man to solve it, by moving only two rods to change it from a dodecahedron to an icosahedron, would have the privilege of spending the night with her. (VOY: "Alliances")
William Chapman was self described as being "not much of a dancer", but thought that perhaps the idea of dancing with Seven of Nine on their date in 2375, "might be fun." (VOY: "Someone To Watch Over Me")