What styles of barbeque exist in American cuisine?

There's a great YouTube video by Rhett & Link called the The BBQ Song (A Review of BBQ in the Southern United States) that is actually a really good starting point for answering this question.

Here are the lyrics to the song:

In the mountains of Tennessee, they like a smoky sauce;
But over there in ol' Memphis, a dry-rubbed rib is boss.

The folks down in Georgia can't make up their mind.
You can give it to 'em sweet or spicy, and they'll say "that tastes alright."

You might think that South Carolinians are just a little bit off.
Would you believe their barbecue's yellow—with a mustard sauce?

Pork shoulder is the cut of choice in Mississippi
And they pride themselves on barbecue that's totally vinegary.

And Louisiana, them Cajuns,
They like to spice it up. They like to spice it up.

This is our review of barbecue,
In the southern United States.
And when my life is through,
Bury me in barbecue,
But make sure it's vinegar-based,
'Cause you know that slows decay,
And it's the style from our home state—North Carolina! (Solo!)

Mutton is big in west Kentucky, that's a fancy name for sheep.
But in the rest of the state, sliced pork butt is what they eat.

Arkansas is kinda like Georgia, a barbecue meltin' pot,
And when you ain't looking, them good ol' boys'll make that barbecue hot!

Alabama's got the strangest thing I've seen in my barbecue days,
'Cause their barbecue sauce is white, made out of mayonnaise!

Down in Texas, where the long-horn steer roam free,
They love to smoke their brisket—that's a big ol' slab of beef.

And down in Florida . . .
Is Florida a southern State?
No, it ain't.

This is our review of barbecue,
In the southern United States.
And when my life is through,
Bury me in barbecue . . .

People not from the South
Think barbecue means "cookout,"
And that's something they're wrong about.

(Spoken outro:)
Barbecue is NOT a verb, barbecue is NOT a grill.
Barbecue is MEAT prepared in a very special way,
Which varies depending on where you go.

You're asking for an entire book worth of information. Rather than trying to answer this myself (heck, I don't even eat meat), I'll give you some references:

• Wikipedia
• Serious Eats (several more articles in that series, search around)
• About.com
• The aforementioned book

There was recently an article in Eater that discusses the various kinds of BBQ which includes the youtube video that is mentioned by Callithumpian:

The traditional American barbecue belt stretches from the Carolinas in the East to Texas and Missouri in the West and from Kentucky in the North down through the deep South. While state lines de-mark significant political and civic parameters, barbecue is not quite so parochial, despite the common stereotype. What we see in the Carolinas for example, are wide swaths of a particular style — most significantly defined by the sauce used — that tend to cross states lines. The simple vinegar and pepper sauce of eastern North Carolina is also popular in the the Eastern part of South Carolina. And similarly, the tomato and vinegar-based sauce of the western Lexington Style bleeds into the Northwestern part of of South Carolina and indeed into Eastern Tennessee and Southern Kentucky.

Source:

Eater, "American Barbeque Style Guide". http://eater.com/archives/2014/07/10/american-barbecue-style-guide.php.