Although often associated with the style of Fred Astaire or Ginger Rogers, the Foxtrot was actually introduced into the mainstream by Harry Fox in 1913 in his Vaudeville Routine. As "Fox's Trot" was embraced by the social dancers of the time, it became simply the Foxtrot.
Foxtrot is a "Ballroom" or Smooth dance, traveling around the line of dance (the perimeter of the room in a counterclockwise direction). "Smooth" is the main description of the dance. Men generally like Foxtrot because it is a lot like walking or strolling. Musically it is very easy to hear the Foxtrot rhythm. The dance is in some ways similar to Waltz, but the focus is much more linear, traveling straight around the room, rather than in a circular fashion. It also has much less of the characteristic "Rise and Fall" of Waltz.
Foxtrot is an extremely useful dance socially and can be danced to a variety of Jazz and Big Band musical styles. International Foxtrot is the very technical style that is commonly seen in Ballroom Dance competitions where couples remain rigidly in the dance hold or "frame." American Style Foxtrot has a fun "theatrical" quality because the couples can open up to allow for spinning the women. Fred Astaire and Gene Kelly both used the long, smooth movements of Foxtrot to cover a lot of ground gracefully in their cinematic routines.
The basic beginner rhythm of Foxtrot is Slow-Slow-Quick-Quick. Higher level patterns are often danced Slow-Quick-Quick. Students have often selected this dance for their First Dance in Weddings.