A private academy in South Korea, i.e. not public schools. Foreigners who teach English in South Korea either work at a public school or a hagwon, though hagwons are not only for language. There are hagwons for all school subjects and a student will attend any kind of one depending on what subject that student's parents want him or her to focus on.
I feel bad for one of my students; she goes to nine hagwons throughout the week. Her parents must be really strict.