(i'm sorry if these links don't show up. blame admin)
Here's a list of the different bending styles and the martial art they're based on. Also, a video demonstrating the style.
-Airbending: based on Ba Gua Kung Fu www.youtube.com/watch?v=CN1fAINfq9o
-Waterbending: based on Tai Chi www.youtube.com/watch?v=TBvF6r6DOvc
-Earthbending: based on Hung Gar www.youtube.com/watch?v=1AwVt0mHXBg
-Firebending: based on Shaolin Kung Fu www.youtube.com/watch?v=3JgMmj4XnEI
There are also other bending styles seen, used by certain characters or group of characters.
-I'm sure we all know Toph uses Southern Praying Mantis style www.youtube.com/watch?v=L0wRqR9J8WA
-Pro-benders use a style reminiscent of boxing from the 1920's. There are differences between each element, but overall they all use the same form.
-We don't see many differences between Northern and Southern forms of waterbending, but from what they show, Northern style focuses more on defense and agricultural methods of bending while Southern style is more offensive.
-The Foggy Swamp Tribe's bending style isn't based on any real life fighting style, but it seems more rigid and straight than northern or southern style, possibly reflecting the stagnant nature of swamp water.