What do Americans call primary and secondary school, college and university?

Primary is generally Kindergarten through 6th grade. Secondary is generally 7th through 12th.

Many cities have K-5th in
"elementary school", 6th-8th in a "middle school" and 9th-12th in "high school", and some places house K-6th as elementary, 7th-9th as "junior high school" and 10th-12th as high. So, in some instances, some might consider 6th also "secondary school." As to college and university, they are often used interchangably. However, most universities are state-funded, rather than privately funded, and almost all universities have accreditation, meaning the credits you earn at one can be transferred to another. Universities have separate "colleges" for different subjects, ie the business college, the teacher's college, etc. Other "colleges", such as community colleges, etc., are often not accredited, and are much smaller, and are also often not state funded. Its kindof confusing. But to put it simply, anyone who is going to an institution of higher learning is going to college. Only some of them are attending a University.

In many areas elementary school still consists of Kindergarten through 8th grade. This is also sometimes referred to as "grammar school" or "grade school".

What was written about community colleges (2 years) in the first post is not true New York State. Every county in the state has a 2 yr community college that is run by the NYS Regents Board. The schools are fully accredited and credits are transferable.