Ballroom
  Smooth - American
    Waltz
    Viennese Waltz
    Foxtrot
    Quickstep
    Tango
    
    
  Rhythm - Latin
    Cha Cha
    Swing (East Coast, West Coast)
      Jitterbug, Lindy Hop
    Rumba
    Samba
  Salsa
    
Modern 

Jazz

Disco (California hustle, bus stop, ...

Folk
  Square
  Contra

Polka

Country Western & Line

Breakdance - Hip Hop

Ballet

Belly

Tap

Clogging

Step (Irish)

Ethnic 
  Greek,  French, Indian

Links:
Just Dance Ballroom : Dance Styles
Types of Dances and Different Dancing Styles | buzzle.com