In the US, school is not standardized. It is controlled at the local level. That means, the names/ages for schools in one town may be different than the names/ages for the schools in a town 5 miles away and both of those may be different than the names/ages for schools in a different state or city. When people in the US move from place to place, they may be in a completely different system. That said, primary/elementary/grammar school tends to be for younger kids. Middle school and Jr. High are common older kids/young teenagers. Older teenagers usually go to high school.

As for the USA, college/university is typically 18-22 year olds; high school is typically 14-18 year olds; middle school: 11-14 year olds; elementary school: 6-11 year olds; kindergarten: anywhere from 4-6 years; pre-kindergarten is typically for play dates for children 5 years and under but does not provide formal education.

Here in the US, we have elementary school which is for students aged 7-11 years old, then there is middle school (or junior high school) which is for students aged 12-14 years old, and then there is high school which is for students aged 15-18 years old. After high school is college or university. At least that's how it was where I lived, and when I used to go to school.