While the image to this day of the Cotillion is elegance, originally the cotillion was a country dance in France. The Cotillion was usually followed by a Quadrille or four couples opposite each other and is rumored to have arisen before the Quadrille.
The Cotillion arose in the West Indies in the early 1700s, but made its way to France by the 1750s. It then made its way to England shortly thereafter, in the 1760s. Around 1844, the Germans were dancing the Cotillion. A notable choreographer, Augustus Lumely, from London, added several variations that are still used to this day. From German, in the late 1800s, the Cotillion reached America and became all the rage. It lost the name cotillion and became simply known as "The German."
In French, the "cotillion" means "petticoat." And this name is derived from the fact that women would raise their petticoats, showing their feet, while dancing, like the Cancan. Of particular note, at the Cotillion, the lady would hold a candle, and if two gentlemen approached the lady, the loser must hold the candle. In short, there were ways in which the men were made ridicule. For example, in on Cotillion, the lady would throw an apron, and the men would pick up the apron, unfolded it, and tied it around their coats. The gentleman who did so the quickest danced with the lady who threw the apron. Many other games were originally played in a coquettish manner.
The Cotillion itself was formed by 8 couples, generally. And the figures would alternate among one another, much like a square dance. With the addition of the Quadrille, more figures were added. Eventually, gifts and prizes were given to dancers and figures.
To this day, the Cotillion is still performed, but mainly at different coming-out balls, debutante balls, charity dances and more. They are to teach etiquette and social manners. But they no longer resemble the old Cotillion. They incorporate Waltzes, Fox-trots, Mambos, and other more contemporary ballroom dances.
Eventually, the Cotillion, Contredanses, Lancers, Hornpipes, and many other dances, would go on to become what we could call "barn dancing" and "square dancing." The ballroom dance would be another thing altogether.