In general, a school is where minors receive an education. There are schools in The Hunger Games, also.
Schools in the districts
Each district's school is geared towards preparing the district children to work in the district's industry. In the poorest districts students attend school until they turn 18.
In District 12's school, every lesson is related to coal, except for basic reading, math and the weekly lecture on the history of Panem. The history of Panem as taught in school is noted by Katniss Everdeen as "mostly a lot of blather" about what the districts owe the Capitol, rather than an actual account of The Dark Days. The school organizes tours of the mines annually. The school also tells students about mandatory television programming.
In other districts, some students and teachers work in the district's industry out of school. In District 8, Bonnie and her teacher Twill worked a four-hour shift at the textile factory, that specialized in Peacekeeper uniforms, after school. In District 11, the school doesn't have lessons during harvest time and everyone, including Rue, worked full-time. However in District 12, students are not allowed to work in the mines as coal miners until they are over the age of 18.
The Capitol has schools as well. In the Catching Fire film, President Snow's granddaughter mentioned that everyone in school is wearing Katniss's braid. It is unknown which subjects are taught in the schools.
District 13's school is called the Education Center. People who are required to attend would have the schedule listed in the temporary tattoo they received each day. Delly Cartwright said that the school is much more interesting than District 12's. Apart from the Education Center, people over the age of 14 are given combat training in preparation for the Second Rebellion.