A dance was a, typically, formal social event designed for couples or individuals to mingle and dance.
In 2269, during the Enterprise's visit to Melkot, which was a recreation of 1873 Tombstone, Arizona, Sylvia shared with Pavel Chekov, who she saw as William Claiborne, that she had just gone shopping in preparation of the upcoming dance that was only one week away. Chekov, who was going along with the scenario, assured her that he was looking forward to it the dance "most eagerly", before she began sharing the contents of her shopping bag, and that the material she had for her dance gown would also make a beautiful wedding gown, and then asked him, "[w]hy don't we just turn that dance into a wedding ball? It would be so heavenly." (TOS: "Spectre of the Gun")
1893-Guinan thought that the prospect of assisting Data would be "more interesting than throwing a tea dance." (TNG: "Time's Arrow, Part II")
The desk sergeant included in the 1940s-era Dixon Hill series holoprogram interacted with Beverly Crusher asking her if she liked Tommy Dorsey, then asked her out on a date, by informing her that he had two tickets for the dance the following night. (TNG: "The Big Goodbye")
Trip Tucker shared the concept of regret with Kov, in 2151, the time of the first dance he had ever went to. He spent weeks practicing the two-step with his brother to be sure he was ready, because he wanted to dance with a girl there he had a crush on, named Melissa Lyles. On the night of the dance, Tucker explained, "[a]ll I wanted to do was ask her to dance with me, but I never worked up the courage to go over and talk to her," adding, "[i]t's been more than twenty years and I'm still kicking myself for not asking that girl to dance." (ENT: "Fusion")
When Kevin Riley, who was under the influence of the polywater intoxication, sealed off engineering and declared himself the ship's captain in 2266, he announced to the crew that "[t]here will be a formal dance in the bowling alley at 1900 hours tonight." Later, when James T. Kirk and Montgomery Scott gained access to engineering, Riley, in moment of disappointment, stated "No dance tonight." (TOS: "The Naked Time")
When Claudius Marcus attempted to impress Kirk while watching gladiator fights, Kirk shared that, "in some parts of the galaxy I have seen forms of entertainment that makes this look like a folk dance." (TOS: "Bread and Circuses")
When Berlinghoff Rasmussen explained how he couldn't help Jean-Luc Picard in the decision he had to make, Picard explained the importance of choices, which in turn allowed the future to be manipulated. He then gave the examples of "Do I ask Adrienne or Suzanne to the spring dance? Do I take my holiday on Corsica or on Risa? A person's life, their future, hinges on each of a thousand choices. Living is making choices." (TNG: "A Matter Of Time")