Here's a little more verbiage I found at word-detective.com. Not a lot of description. And, incidentally, Begin the Beguine, the Cole Porter song, is a foxtrot!

And, btw, no luck on finding a video. I'll keep looing though. The dancelessons web page is cool, but it has no search engine, so you have to page through everything.
The "Beguine" referred to in the title of Porter's hit song of the 1930s is, as the Oxford English Dictionary puts it, "a kind of popular dance, originally associated with Martinique; also applied to a kind of syncopated dance rhythm." Of course, anyone attempting to actually learn to dance from that definition is going to have serious social problems, but I understand the beguine is a ballroom dance similar to the rumba. The name "beguine" comes from "beguin," French slang for "infatuation." (Martinique, a West Indian island, was once a French colony.)
For a word that became associated with a highly romantic dance step, "beguin" had a remarkable beginning. The Sisters of Beguine were a 12th century Catholic lay order founded in the Netherlands by Lambert Begue. While the Beguines were suitably pious, their vows were not as strict as many other orders, and the Sisters were allowed to leave the order to marry if they wished. Perhaps because of this freedom, "beguine" came to be used as a slang term for "flirtation" or "infatuation." The Sisters of Beguine still exist in small communities in the Netherlands, but, to my knowledge, they do not dance the beguine.