Ballroom
  Smooth - American
    Waltz
    Viennese Waltz
    Foxtrot
    Quickstep
    Tango
    
    
  Rhythm - Latin
    Cha Cha
    Swing (East Coast, West Coast)
      Jitterbug, Lindy Hop
    Rumba
    Samba
    Tango
    Salsa
    Merengue
    Flamenco
    Reggae
    
Modern / Contemporary - Centers on a dancer's own interpretations instead of structured steps

Ballet

Jazz

Lyrical - Fusion of ballet with jazz

Folk
  Square
  
  Contra - Contra dance refers to several partnered folk dance styles in which
  couples dance in two facing lines of indefinite length.  A fundamental aspect 
  of contra dancing is that the same dance, one time through which lasts roughly 
  30 seconds, is repeated over and over-but each time you dance with new neighbors.
  It has mixed origins from English country dance and French dance styles in the 17th century.
   Sometimes described as New England folk dance.
  The New England folk revival in the 1960s, brought brought it back to the mainstream.
           
          
  Ethnic 
    Indian
    Scottish Country Dancing
    Scottish Highland Dancing
    Irish Jig
    English - 
    Greek
    Turkish
  Clogging  
  
Historical
  Medieval
  Renaissance
  Baroque
  
Country
   Overlaps with contemporary folk dance and ballroom dance
    Western & Line
    English Country
    
Swing
  Modern Jive
  Jitterbug
  Lindy Hop
  Jive
  Western
Polka

"Street Dance" or "Vernacular Dance" 
   hip-hop
   Disco (California hustle, bus stop, ...
   Breaking
Pop

Pogo - Punk Rock Pogo

Breakdance - Hip Hop

Belly

Tap

Step (Irish)

Links:
Dance Types - Styles of Dancing | about.com
List of dance style categories - Wikipedia
List of dances - Wikipedia
Types of Dances and Different Dancing Styles | buzzle.com
Just Dance Ballroom : Dance Styles
Contra dance - Wikipedia Folk dance - Wikipedia Dance, Performance and Entertainment Arts Resources