There are many forms of swing dance - and there is a reasonable amount of disagreement in the dancing community about what is and isn't a swing dance. Names also vary, one dance may be known by more than one name, another name may have been applied to different dances.
Probably the big three at the moment are:
Lindy Hop (sometimes also called Jitterbug), one of the first swing dances which came out of the ballrooms of Harlem in the late 1920's and early 1930's.
East Coast Swing (also known as Eastern Swing or 6 beat swing), is basically a cut down version of Lindy Hop with some ballroom influence.
West Coast Swing (also known as Western Swing - though Western Swing can refer to another dance style as well), has evolved significantly from its roots and is now danced to modern music.
There is also:
Balboa (also known as 'Bal' and split into Pure Bal and Bal-Swing)
Charleston (the dance that came before Lindy Hop)
Jive (the ballroom version of swing dancing)
Shag (many different varieties including Collegiate Shag, Carolina Shag, St. Louis Shag, Heel Shag.....)
Rock and Roll and Rockabilly (not really swing dances, but they came from swing dancing - same goes for Modern Jive and Ceroc).
Blues Dancing (also not a swing dance, but is danced by many swing dancers).
There are also a variety of other forms that were only danced for a short period, in a particular location and have been largely lost, or were not part of the swing dance revival.