American schools from kindergarten to high school (which is our word for the last thing before going to a college/university) mostly divide the school year into quarters, which just means literally what the word means: one of four parts. So a quarter is a half of a semester. That's how often official grades are recorded, but it makes almost no difference in terms of schedules. Most classes span the whole school year, with just one spot (in a schedule of 6-8 classes) rotating from one class to another when one quarter or semester ends and the next begins.

But, like I said, that's before college. In American colleges/universities, quarters disappear completely and and classes are scheduled by semester only. Even if you go to the same place at the same time each day to study the same subject continuously for a whole school year, it's officially two independent classes, one for each semester. But the amount of education you need for a degree or have taken so far, and the weight of a given class in your grade average, is not measured in semesters. It's measured in "credit hours", which refers to how many hours are spent in class per week. Most classes are 3 credit hours, but they can be 2-5 (with 4 and 5 usually being for classes with both a 2-3-hour lecture component and a 1-2-hour lab component).

You might be allowed to take some classes without pursuing a degree, but each department will also have some classes, usually the more advanced ones, that you can't take unless you have declared that field as your "major" and/or you've taken certain prerequisite other classes first (or taken and passed the prerequisite's final exam to skip the prerequisite).