Standardized Ballroom dancing comes in two styles, American and International.
American style has a basically different development, contributed to by many sources - U.S. studio chains such as Arthur Murray’s and Fred Astaire, and independent U.S. studios and exists in some level of standardized form in each. Several certifying organizations exist and the curriculums can generally be quite variable between chains and studios. Organizations such as the National Dance Council of America (NDCA), the United States Imperial Society of Teachers of Dancing (USISTD) and Dance Vision (DVIDA), among others, offer standardized dance syllabi in the American Style.
The International style was developed by the British, particularly through the Imperial Society of Teachers of Dancing (ISTD) and the International Dance Teachers' Association (IDTA), and are also the certifying organizations of this style, additionally to some others. This style has the benefit of being both less random in terms of studying its syllabi as well as more technical – and is used by many to develop strong form and technique for ballroom dancing as a whole (including American). While most of the world uses international style exclusively, both styles are popular in North America.
American Smooth versus International Standard
Each style has specific dances that are categorized as either American Smooth and International Standard. American Style includes primarily the Waltz, Foxtrot, Tango, Viennese Waltz and may also include Peabody and in some studios, Argentine Tango (some consider this a Rhythm dance), whereas the International styles keeps it's own version of Waltz, Foxtrot, Tango, and Viennese Waltz, and also adds Quickstep.
Although American style smooth and international style analog are taught very differently, the styles are very similar. The most obvious difference is that int'l standard includes quickstep, which is not part of American smooth. The other major difference is that int'l style permits figures in closed position only, while American style allows open positions and even solo actions. Beyond that, the main differences between the two styles are in emphasis rather than result.
American Rhythm versus International Latin (or Latin American)
Compared to smooth/standard, rhythm/Latin has relatively little overlap. While each category has cha cha, rumba, samba and a swing dance; International has paso doble, while American has bolero -- another, slower rumba dance -- and mambo. Most importantly, the hip motion differs: in the American style the characteristic hip action is “delayed”, whereas the International is not. This gives to a rhythmical hip driven appearance in the American, and a powerful body driven appearance in the Latin. One should know both skills to successfully execute the variety of open choreography in both.
Use in Teaching
In my own teaching, I tend to integrate aspects to make a more cohesive whole, in the beginning with students or those that come to me from other instructors. There are fundamentals that I use from International Standard dance that make learning American Smooth much easier in the beginning, whereas there is a technique in International Latin American dance that makes the characteristic “delayed hip response” in American much easier to learn, in my experience teaching for over 20 years. In this way, I can give my students a foundation that allows them to naturally evolve to whatever style of ballroom they prefer, and allows them to enjoy all that ballroom has to offer.