Well, there's a wide variety of "American" style.
First two main distinctions: NY/NJ style, which is generally the traditional American pizza. 1" crust, a little crispy, thin body, sauce, lots of cheese. This has a few variations though, most notable "thin crust", which is often cracker-thin and very firm & crispy, the body much thinner too.
The other main variety is Chicago deep-dish pizza, which is very thick, anywhere from 1-3 inches deep in my experience.
Both of these in their pure forms are difficult to find outside of their regions. But the NY/NJ style is popping up in places around the country. North Carolina Charlotte area for example, as transplants from the North-East find their way to other parts of the country.
Then there's the various pizza chains: Pizza Hut, Dominos, Papa Johns... most of their pizzas most closely resemble the NY/NJ, but that's putting it loosely. They're sort of like a McDonald's Big Mac compared to a steak house burger, except at least for me, I sometimes just want the Big Mac, but I'd never choose the chain pizza when the alternative is available.
Let me be clear though: some people are purists to one type or another and consider other varieties, or chain pizzas, to be blasphemy. But if you put aside ideological considerations, almost all pizza is fundamentally good. You can find bad pizza, but it's harder than finding good pizza. (Let the downvotes roll in from Chicago/NY purists :D )