The foxtrot is a ballroom dance.

It is often said that foxtrot took its name from its inventor, the vaudeville actor Harry Fox; however the exact origins are unclear. The dance was premiered in 1914, quickly catching the eye of the talented husband and wife duo Vernon and Irene Castle, who lent the dance its signature grace and style.
W.C. Handy ("Father of the Blues") notes in his autobiography that Noble Sissle told a story that Handy's Memphis Blues was the inspiration for the Fox Trot. Jim Europe, the Castle's music director, would play slowly the Memphis Blues during breaks from the fast paced Castle Walk and One-step. The Castles were intrigued by the rhythm and Jim asked why they didn't create a slow dance to go with it. The Castles introduced the "Bunny Hug" in a magazine article. They went abroad and in mid-ocean sent a wireless to the magazine to change the "Bunny Hug" to the "Foxtrot."
It was later standardized by Arthur Murray, in whose version it began to imitate the positions of Tango.
At its inception, the Foxtrot was originally danced to ragtime. Today, the dance is customarily accompanied by the same big band music to which swing is also danced.
From the late teens through the 1940s, the foxtrot was certainly the most popular fast dance and the vast majority of records issued during these years were foxtrots. The waltz and tango, while popular, never overtook the foxtrot. (Even the popularity of the lindy hop in the 1940s didn't dent the foxtrot because the foxtrot could be danced to those lindy hop records, as well.)
When rock and roll first emerged in the early 1950s, record companies were uncertain as to what style of dance would be most applicable to the music. Famously, Decca Records initially labelled its rock and roll releases as "Fox trots", most notably "Rock Around the Clock" by Bill Haley and His Comets. Since that recording, by some estimates, went on to sell more than 25 million copies, "Rock Around the Clock" is technically the biggest-selling "Foxtrot" of all time.
Over time, Foxtrot split into slow (Foxtrot) and quick (Quickstep) versions. In the slow category, further distinctions exist between the International or English style of foxtrot and the continuity American style, both built around a slow-quick-quick rhythm at the slowest tempo, and the social American style using a slow-slow-quick-quick rhythm at a somewhat faster pace.
In the context of International Standard category of ballroom dances, for some time Foxtrot was called Slow Foxtrot, or Slowfox. These names are still in use, to distinguish from other types of Foxtrot.