"Hagwons" (학원, 學院) are the Korean term for a learning institute or academy. They teach computer, math, taekwondo, English and much more. They are after-school academies, commonly used for Korean youth, but also for university students and adults. They are everywhere, in all parts of every town and city in Korea, and most Koreans have attended at least one at some point in their lives.
All in all, 'hagwons' are the most likely choice for first time teachers deciding to come to Korea.
Now specifically, what can you expect out of teaching at hagwons in Korea? Well, in general terms, here it is:
A typical classroom at an English hagwon.
You must be prepared to leave behind some preconceived notions about teaching and about Asian students and Asian society as we know it in our home country. Your students will often not be sons and daughters of rich parents who have their kids get a western education. Those Koreans and Chinese back home are highly motivated to succeed in school because of the Confucian concept of duty. The students you will have are an average cross-section of society have been in school for 6 or 8 hours by the time they get to your hagwon, and are up to here with studying already. But this is expected by their parents and very few students play hooky. However, you will get a little sense of how important student enrollment is to the hagwon when the owner asks you to change little things about your appearance or behaviour which may have nothing to do with being a good teacher. The public image the hagwon displays is just as important to the owners as the education, because it is a business.
That being said, it also equally important to be flexible as a teacher in timetabling and scheduling, because Korea is the country of sudden changes and adjustments. Usually, no notice is given. It is both a language issue, where every Korean who works at the hagwon may know about the change in advance, except you, AND it is simply Korean culture to be very flexible in timetabling, especially in smaller businesses, which hagwons usually are.
Some English hagwons are huge and employ up to 20 foreign teachers, however most tend to have between 2-5 foreigners and about equal or double the number of Korean teachers plus a secretary and the hagwon manager. So organization, staff meetings and such are not a given, although some hagwons do have them.
For more info on checking out your school before you commit, make sure you read the article on qualifying schools.