The terms syncopation and syncopated step in dancing

Dance

Dance is an art form that generally refers to movement of the body, usually rhythmic and to music, used as a form of expression, social interaction or presented in a spiritual or performance setting....

 are used in two senses:

1. The first one matches the musical one
   Syncopation
   In music, syncopation includes a variety of rhythms which are in some way unexpected in that they deviate from the strict succession of regularly spaced strong and weak but also powerful beats in a meter . These include a stress on a normally unstressed beat or a rest where one would normally be...
   
   : stepping on (or otherwise emphasizing) an unstressed beat
   Beat (music)
   The beat is the basic unit of time in music, the pulse of the mensural level . In popular use, the beat can refer to a variety of related concepts including: tempo, meter, rhythm and groove...
   
   . For example, ballroom
   Ballroom dance
   Ballroom dance refers to a set of partner dances, which are enjoyed both socially and competitively around the world. Because of its performance and entertainment aspects, ballroom dance is also widely enjoyed on stage, film, and television....
   
    Cha cha
   Cha-cha-cha (dance)
   The Cha-cha-cha is the name of a dance of Cuban origin.It is danced to the music of the same name introduced by Cuban composer and violinist Enrique Jorrín in 1953...
   
    is a syncopated dance in this sense, because the basic step "breaks on two." When dancing to the disparate threads contained within the music, hands, torso, and head can independently move in relation to a thread, creating a fluidly syncopated performance of the music.
2. The word "syncopation" is often used by dance teachers to mean improvised or rehearsed execution of step patterns that have more rhythmical nuances than "standard" step patterns. It takes advanced dancing skill to dance syncopations in this sense. Advanced dancing of West Coast Swing
   West Coast Swing
   West Coast Swing is a partner dance with roots in Lindy Hop. It is characterized by a distinctive elastic look that results from its basic extension-compression technique of partner connection, and is danced primarily in a slotted area on the dance floor...
   
    makes heavy use of "syncopation" in this sense (although swing music and swing dances feature the "usual" syncopation, i.e., emphasising the even beats).

Many dance teachers criticize the use of the term "syncopation" and abandon it in favour of the term "double-time". This is most likely due to a convenience in similarity, and/or a misunderstanding of the rhythmic concept.

Dance syncopation often matches musical syncopation, such as when (in West Coast Swing) the leader touches slightly before beat 3 or stomps on beat 6.