The first decades of the 1800s saw increasing departure from the formal dance routines of the earlier century and the introduction of livelier and more intricate routines. Many manuals were published to educate dancers in new and old steps. Thomas Wilson, who wrote several dancing manuals, created an overwhelming diagram of possible Quadrille formations, to illustrate one of his books. This link will take you to several interesting original manuals.
One of the concerns with dancing was that a lady should not dance with anyone before a proper introduction. This did not mean, "Hi, I'm Josh. Wanna dance?". A mutual acquaintance must formally introduce a lady to the gentleman. This posed a problem during country dances and quadrilles, as an introduced couple would be continually changing partners during the course of the dance, and therefore, dancing with who knows who. Or whom.
The waltz was introduced in England during Regency times, much to the confusion and dismay of some. While other dances had exuberant moves and physical contact between the sexes, the waltz outdid them all. One couple danced exclusively with each other, with both arms in contact for a whole dance set --imagine the possible outcomes! Despite the naysayers, the waltz was very popular and three-quarter time ruled for the rest of the century. Other lively couple dances were the mazurka, which was usually danced by couples in quadrille formation, the galop, and the polka. (I do hope you watched that last link!) All of these were descended from European folk dances.
Especially toward the mid-1800s, etiquette and decorum at dances was taken very seriously. To avoid awkward scenes, men who wished to dance with a certain lady requested a dance later in the evening. She would then write his name on her dance card, which was usually marked with a list of the type of dance music that would be played. Often the gentleman would request her honor of dancing a waltz, or whatever dance he preferred. When that set approached, he would then approach the lady.
I'm finding it difficult to locate decent pictures of men's fashion from Regency era publications. If the clothing is fine, the faces are usually just awful. Ackermann's women are so lovely, I can't bear to pair them with poorly-drawn men! What I've done with some is to take the bodies and give them new heads, usually from my file of handsome men in daguerreotypes. Those of you who visit My Daguerreotype Boyfriend know that good-looking men weren't invented yesterday. Well, Minerva meets a handsome stranger who is living just a bit in the future. Who is he? Hint: He's got a famous dad and looks absolutely nothing like him.