Public School - supported by taxes
Private School - supported by fees paid by students' parents
Parochial School - a religious, private school
Vocational School (Trade School) - these are usually post-secondary schools, but in some cases take the place of the final years of high school.

Middle School - In some school districts, there is a middle school instead of a junior high school. Middle schools may include grades 6-9, grades 6-8, grades 7-9, etc. Whatever is convenient for the school district at that time.

Recess - time when elementary school students leave the classroom and go out to play in the school yard.

Elementary school (and its synonyms) can refer to grades 1-6 or grades 1-8.
Students who graduate from high school receive a diploma.
GED (General Equivalency Diploma) - Students that did not finish high school can take the GED Test and receive a General Equivalency Diploma, which is equivalent to a high school degree.
K-12 education - a short way to refer to kindergarten through 12th grade education.