The dances were intricate and took quite a while to learn, which is why children were taught to dance as rigorously as they were taught to read. Dances were also done in "set" meaning sets of two, so if a lady accepted a dance from a gentleman, she was his partner for two dances.
There were many dances done during the Regency. The quadrille was a popular dance involving four couples dancing together and making intricate patterns together on the dance floor.
Christina Guza, one of my colleagues on the Beau Monde, a Regency chapter of Romance Writers of America, found this wonderful video demonstrating the Quadrille.
As you can see, everyone needed to know exactly what to do and when to do it, or it threw off everyone else. It reminds me old fashioned square dancing, in a way, but much more complicated, and there's no "caller" to tell dancers what to do next.