Kindergarten-6th grade, Elementary school (ages ~5-12): general childhood education in all subjects (spelling/reading, math, history, etc.). Each year, everybody is placed in a class with ~30 other students, with one teacher for the whole class teaching everything. Grades shouldn't matter, so long as you pass.
7th and 8th grades, Middle school (ages ~12-14): each subject has a different teacher, and instead of being with the same class all day, each student sees each teacher for a fraction of each day, so each period a teacher has will have different students in it, but every student has to take the class at some point in the day. Each student does have to take specific core classes no matter what (English, math, social studies, science, and gym, generally), and fill in the remainder of their days with elective classes that they choose themselves (music, art, etc.). Grades still don't really matter, here. (it's probably redundant to explain how periods and core/elective classes work, but I'm honestly not sure if this is how UK schools do it, too)
9th-12th grades, High school (ages ~14-18): exact same as middle school, in terms of how your schedule works and core/elective classes. This is where it gets confusing, though. In order to graduate, you have to have taken enough of each type of class by the end of 12th grade. For instance, generally, you'll need to have taken English every year of high school. Also, four years of social studies, three years of math, three years of science, two years of a foreign language, one year of both art/music and physical education, and I guess you can fill up the rest with whatever you want (these year-requirements are just general, though, and will vary depending on your high school). Anywho, each year ends up being worth one credit, and you get a grade each semester (half a school year) and generally you'll need from somewhere between 21-26 credits (again, dependent on your school) total to graduate, including all the required ones. And that is the culmination of your public education experience. Going beyond is entirely optional (technically, so is graduating high school, but whatever)
College/university/higher education/whatever you want to call it is, for all intents and purposes, however, entirely separate. It's no longer part of the public schooling system, so it does get complicated, and if anyone's read this far, they should probably stop unless they really want to know for some reason. Students have to apply for college by themselves, and to decide whether or not you get in, the college will look at things like your high school coursework and GPA, extracurricular activities, and SAT/ACT scores. Grades work, of course, by points and percentage (I really hope that is how the UK does it, too). 90% or higher is an A (4.0 GPA), 80-89% is a B (3.0-3.9), 70-79% is a C (2.0-2.9), 60-69% is a D (1.0-1.9), and 59% or lower is an F (0), failing. Also, you can get, for example, a B- for 80-83, and a B+ for 87-89, and so on for each letter, but no one cares. Colleges will average out your GPA and other bullshit and use it to compare you with other students, to see who they want to admit or not. They also use the SAT or ACT to compare students, which are voluntary tests you sign up and pay to take, to give colleges more reason to deny you admission. SATs have three sections, reading, math, and writing skills, and you get 200-800 in each section, giving you a total score of 600-2400. ACTs have four sections, English, math, reading, and science, each which has a score from 2-36, and the sections are averaged out, to give you a cumulative score from 2-36. The tests are really just to compare you to other applicants.
And of course, there's all the other crap about college, like old people applying, or how colleges use all of your admission stuff to place you into programs, and graduating from college itself, but that is quite impossible to explain, since literally every college in the country will handle things differently.
And this is nowhere near abridged, and probably does the opposite of what I wanted, by confusing everything more. Even though I think it's not terribly complicated.
tl;dr- okay, so it is confusing.