The Term American style refers to a particular style of Ballroom Dances developed in America.
The American Style is made up of two categories of dance
American Smooth 
Waltz, Tango, Foxtrot and Viennese Waltz
and American Rhythm
Cha Cha, Rumba, East Coast Swing, Mambo and Bolero

American Rhythm loosely corresponds to the Latin category of the International Style The American Rhythm dances were also influenced by American dancers of the post-WWII era, but the differences between this style and International Latin are more subtle. While step patterns are not identical, movements are as free and explicitly sensual as in International Latin.

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Cha Cha

The Cha-Cha (originally known as the Cha-Cha-Cha) is a fun & flirty dance that grew out of the Mambo & Rumba. Originated in Cuba this dance became immensely popular in the United States in the late 1950’s.

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Rumba

American Style Rumba is commonly danced in the United States, with box-like basic figures. The Rumba spotlights the lady’s rhythmic body movements and hip actions resulting in intense, almost steamy, scenes of passion.

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East Coast Swing

The Swing is a “must learn” dance. Based on the Lindy Hop this dance was created by dance studios in the 1940s. Lindy Hop was felt by dance studios to be both too difficult and too unstructured to teach to beginning dancers, but there was market demand for training in Swing Dance.

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Bolero

Bolero is one of the competition dances in American Rhythm ballroom dance category. It is a slow dance characterized by smooth, gliding movement, dramatic arm styling and a romantic feel.

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Mambo

Mambo is a Latin dance of Cuba, which was developed in the 1940s when the music genre of the same name became popular throughout Latin America. Later on it became the most fashionable Latin dance in the United States.