Lindy Hop was named after Charles Lindbergh's flight to Paris in 1927, when the newspaper headline read: "LINDY HOPS THE ATLANTIC."  The dance itself has no "hop" in it.  It's smooth and solid with a steady 8-count beat.  In the late 1920's, Lindy Hop was the dance to do, but didn't get its reputation until the Savoy Ballroom opened in Harlem.  At the Savoy, Lindy got hotter and hotter with top bands performing for the dancers.  It also got better and better as good dancers became great dancers through popular competitions on the weekends.  The dance wasn't just fun to do, but also fun to watch as this period saw the creation of steps and styling constantly.  At the peak of Lindy Hop's meteoric rise, Frankie Manning created the first airsteps in 1935 and the dance soared.

Lindy Hop is a social dance.  Partners are connected smoothly and gently to each other, while relating closely to the music, in feeling, improvisation and phrasing.  Many films show aerials, often done at very fast musical tempos.  They require a superb degree of expertise and are not danced socially, but only for performance.  Aerials, or "airsteps", are impressive and spectacular, so that's what you see in the movies!

Lindy Hop has its roots in both African and European dance, with the embracing hold and the turns from Europe, the breakaway and solid, earthy body posture from Africa.  The dance evolved along with the new swing music, based on earlier dances.  It continues to evolve today, with a few different styles of lindy including Savoy and smooth styles.  CLIPS

http://www.savoystyle.com/history.html

http://dancing.org/lindy-what-is.html
      
This is sometimes referred to as the Jitterbug.  The original Jitterbug was Lindy and West Coast style dancing, but many membership dance studios wouldn't teach Lindy, thinking it a dance of the economically disadvantaged.  However, independent studios and dance halls were garnering more crowds by teaching Lindy, and so the old dance association that all of the other studios were members of decided to "refine its cavorting".  So they changed it, and called it Jitterbug or Lindy.  However, this was not the dance that they were truly doing, but newer dancers were never taught the difference.  Eventually the "refined" dance became what we know today to be East Coast Swing.  The term Jitterbug migrated with it, though Lindy and West Coast is still sometimes referred to as Jitterbug.

This dance is easier to teach and easier to learn, which is why many organizations will teach it first.  It creates a good foundation for learning the other, more complex swing dances detailed on this site.

More detailed history: http://www.streetswing.com/histmain/z3ecs1.htm
        
Balboa (or Bal) was created in, and named for, the Balboa Park Club, an enormous wood-frame building that was home to the Naval Officer's Club in San Diego during WWII.  The Balboa has lots of fast footwork but not much whole-body movement, and was developed as an answer to the question, "what do you dance when the band plays Dixieland jazz?" (The Balboa uses a "shuffle" basic, with short foot movements and overall "look" optimized for really fast swing music, such as Dixieland)  It's extremely useful for very crowded dance floors and high tempos, though it can also be danced at slower tempos.   CLIP

http://www.swingcraze.com/ussds/other_swing_dance_styles.html
        

Blues is the name for an informal type of swing dancing with no fixed patterns and a heavy focus on connection, sensuality and improvisation, often with body contact.  Although usually done to blues music, it can be done to any slow tempo 4/4 music, including rock ballads and "club" music.  Blues can seem a little too close due to the close connection and body movements, but in actuality blues can be done with a fair amount of space between the dancers as long as connection and frame remain intact.  It is an awesome replacement for the ol' bump and grind at the club, if only just to change things up.  CLIP

http://en.wikipedia.org/wiki/Blues
        
The Charleston is a dance, named for the city of Charleston, South Carolina. It was popular in the 1920s, and spawned Lindy Hop in the 1930s. Charleston is danced in 8-count: solo, with a partner, or in groups (usually facing lines).  It is heavily influenced by traditional African dancing and borrows its attitude from the original Cakewalk dancing that African slaves would do on plantations.  CLIP

http://en.wikipedia.org/wiki/Charleston_%28dance%29

20's Charleston is usually associated with flappers and ragtime, but is a versatile 8-count dance.  Lindy Hop, Shag and Balboa are acknowledged to have come from this zany little Flapper dance.
With small, twisting footwork and a back and forth motion, it's easy to see why so many dancers used Charleston and adapted it with kicks for use in swing.

For a time, Charleston and dances related to it were banned from dance halls like the Savoy because there just wasn't enough room for the kick variations that were so popular.

http://www.jungleswing.ca/classes/charleston.html
       
Collegiate Shag
This dance was extremely popular on the West Coast, where it was used in dace competitions such as The Harvest Moon Ball. The basic consists of a 6 count pattern: slow (2 counts), slow (2 counts), quick, quick (1 count each.) Today the dance has been taken to a new level by the addtition of aerials and intricate footwork patterns.

Carolina Shag
Born in the 1950's around the Myrtle Beach area, this style is still popular today in the Southeastern part of the U.S. In fact, it is South Carolina's State Dance. It consists of 6 and 8 count patterns. It has the appearance of someone doing West Coast Swing with rubber legs and tight footwork.

St. Louis Shag
Is an 8 beat style of Shag which can be done at very high speeds. The dance is most comforatable at a tempo of 200-300 beats per minute. It is an offshoot of the original Charleston dancing. The basic patterns are close together with no basic step patterns such as you see in West Coast Swing.

http://www.swingdancers.com/shag.asp
When Jitterbug was banned from virtually every serious dance hall in the late '40s "Sophisticated Swing" began to flourish.  Current WCS styles vary considerably.  Modern WCS can be conservatively upright-postured, smooth and warm, or a funky, hot partnered-jazz dance.  A bit more self-conscious than the wild abandonment of '40s jitterbug, one of the first generic names for WCS was Sophisticated Swing.  West Coast Swing is the state dance of California.  It's origins are definitely in Lindy Hop, but it has really come into its own since its creation and introduction.

http://people.cornell.edu/pages/kpl5/Link15_wcs.html