If you would like to teach in Korea, too, you have some decisions to make. Aside from where in Korea you would like to teach and when you want to leave on your teaching adventure, deciding between hagwons (private schools) and public schools is a big first step.
From what I have heard from others who have gone to teach in South Korea, there are advantages and disadvantages to each type of school. Hagwons usually have much smaller class sizes than public schools. Think 15 students versus 40. Hagwons may also have only evening classes, so you will want to know what you are getting yourself into. The way a hagwon is run depends on the owner, so you may have a joyous time abroad or want to go home as soon as you set foot in the place. Stories abound of teachers who have not been paid or of owners who treat their employees very poorly. My advice: Do your research of a hagwon before signing any contracts.
I decided to go the public school route. Yes, there are more students and the pay might be slightly less than a hagwon; however, my contract is through the government which reassures me that I will get paid with regulated hours. The agency I will be working for is called SMOE (Seoul Metropolitan Office of Education). It places teachers in public schools throughout Seoul.
A similar agency, EPIK (English Program in Korea), places English teachers in Busan, South Korea.
Other government-sponsored English teaching opportunities: