The Foxtrot was popularized by the vaudeville comedian named Harry Fox.  It was one of the most popular dances during the early 1900's when it was presented in the Ziegfield Follies.  The forerunners to the Foxtrot were dances like the Turkey Trot, the Castle Walk, and the One Step.  It was first refined to its present day style by Vernon and Irene Castle.  The Castle’s were credited with saving social dancing from extinction. 

The Foxtrot should be smooth and flowing, with the slows and quicks blending seamlessly into one another.  By slowly controlling the footwork of the directional movements, the dancers’ body’s travel continuously around the floor with grace and ease.  There are two major styles of the Foxtrot.  The American or social style and the International style.  American styles allow for open work and utilize more rhythmical expression is less strict.  The International style is much more exacting, like a ‘science” and requires more attention to details.  As the name implies, the American style is more useful on a social floor unless the dancers are well versed in advanced floor crafting.
 

Flowing, gliding, sustained smoothness.

Progression around L.O.D.

Sustained smoothness, minimal rise & fall

Footwork:

Forward Steps - Slow Counts are:
                      HEEL /TOE  (HT)

Backward Steps - Slow Counts:

1.  Followed by another back step are
                 TOE/HEEL  (TH)

2.  Followed by a side step are  
                TOE/HEEL/TOE   (THT)

 *  1st QUICKS' are TOE   (T)
*   Final QUICKS' in a pattern are
                      TOE/HEEL  (TH)

Note 1.  Most steps between the "1" beat and the "4" beat are always toes, unless using a rotation, like a Impetus Turn, Pivots, etc.

 Note 2. The heel on the last step of a component generally does not lower until commencing the next step.

Tempo: 

American:  30 - 34 MPM
International:  28  MPM

American Foxtrot uses counts of 4, 6 or 8 beats for most variations

         s         s        q      q 
      1 - 2    3 - 4     5      6   

      s      q    q       s      q     q
   1 - 2    3    4    5 - 6    7    8

   s= slow = 2 beats of music
   q = quick = 1 beat of music

There are numerous counts that can be utilized to create advance elements. 

The more advanced dancer use more "&" counts and syncopates steps.  The use of "picture steps" allows holding actions to create more flavor to the dance.  The holding actions sustain a position over one or more measures.