Hagwon (also known as Hakwon or Hogwan) are the most likely place you'll find work. They are private institutions offering children extra classes to keep up with the highly competitive educational atmosphere in Korea. English language hagwons will also often offer classes to adults.
Martin Tilney posted some great information about teaching in Korea on the British Council website.
"The reality is most hagwons place more importance on the image of the teacher than his or her credentials. For example, I am an adopted Korean. Though I'm a native speaker, I look like a Korean. This fact made it a little difficult for me to find a job at first.During the phone interview the recruting agent sounded so excited, but after meeting me in person, they didn't bother to call me again. Unfortunately, in many cases, it is even harder for Indian or African people to get teaching jobs, even though their English is perfect and they are more qualified for the job than Caucasians."
