My limited experience with school in the usa.... kindergarten is provided free of charge for 5 year olds. 4 year olds have preschool but it is not compulsory or available everywhere. Some small town do not have any school before kindergarten, some areas have preschool programs only for those who qualify ( lower income or English is not the homes first language ... those children less likely to succeed without the added instruction. Areas with larger populations usually have more options including prescool at cost to the parents. 3 school is provided to 3 year old with the same options as preschool only with less areas offering 3 schools. 2 years is optional daycare providers who may or may not have learning opportunities gor children at cost to parents. As far as i know there are no other names for the schools but each classroom (in larger areas may use names for the kids in each classroom like owls or unicorns to help the children remember which room or lockers are theres.