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.