One basic distinction among dances (in the modern era) is between improvised, informal dance, on the one hand, which takes no formal training and is mostly a form of courting or Recreation; and, on the other, Ritual? forms of dance which have been formalized and refined into specific genres. (In earlier eras, i.e. before the 1950s, many forms of courting and recreational dance were quite formalized and refined.)
These genres include:
Ballroom Dance -- Belly Dance -- Swing Dance -- Jazz Dance -- [Modern Dance]? -- Folk Dance -- [Scottish Country Dance]? -- Morris Dance -- [Ballet Dance]? -- [Square Dance]? -- Tap Dance -- [Free-Form Spontaneous Dance]?, like [Barefoot Boogie]? and [Dance Jam]?
Historical dance forms: Medieval dance --Renaissance Dance -- Baroque dance -- [Colonial American dance]? -- Regency dance -- Vintage dance