This is a formal, group dance from the early 20th century performed by four couples. It was popular in Europe and in the Gilded Age of the late 1800s and early 1900s in the United States.
Here’s a video.
There’s a very regimented series of moves, with a great deal of “partner passing” and theatrics. In places, it appears to be the classic ballroom dance you would see in Cinderella or similar stories.
Why I Looked It Up
I was reading a book about the Vanderbilts and the Gilded Age. The quadrille was a necessary skill to be accepted among the upper class.