Like I have no friggin clue how the US school system works. "Highschool" is in there somewhere, I think. That one gets mentioned a *lot*. Not sure what would be the german equivalent, if there is even one.
The US school system is divided into 3 "stages" (4 if you count university): elementary school running from kindergarten to fifth or sixth grade (5-11/12), middle school or "junior high" running from 6th/7th to 8th/9th (11/12-14/15), and then high school from 9/10-12 (14/15-18). Many school systems now offer "pre-K" programs for younger kids, especially those from poor families, which are encouraged to help give those kids a better start in school.
High school is most like the German Gymnasium. Unlike in Germany, though, ALL American students go to high school, so a lot of high schools will incorporate programs that are more similar to a trade school. I can't speak for every state, but Texas's intent is to have every student graduate high school ready to go to university, junior college, AND a career. (Any given student would pick one of those options, but they would have all three to choose from.) All schools require every student to take English, science, social studies (history and civics, most schools include geography), mathematics, and a foreign language. They'll also offer elective classes, including arts, trade classes (for example, my school offers welding), home classes (such as child development or cooking), etc.
However, high schools also offer a lot of extracurricular activities - sports being the most obvious, but also drama, band, academic competitions, student council, minority advocacy groups, religious groups sometimes, at a big enough school whatever you can think of they probably have a club for.
My third year of high school, I worked out that between class, academic competitions, and band, I was at the school for ten hours a day or more, and then would usually have a couple hours of homework each night. A typical school schedule will have between six and eight classes that you go to every day, or sometimes you'll have six to eight classes that you go to three/four one day and the others the next.