Find a school that will hire you. Most schools look for native English teachers and there are many options all over South Korea. A good website with a reliable database of different teaching opportunities is goabroad.com. There are usually two different kinds of opportunities to teach English, depending on your education level. Hogwans are after school academies where students go to learn English, among other things and you usually only need a bachelor's to teach. Universities and elementary schools are harder to find openings and usually require you to have completed your master's.
Apply for the proper employment visa. To work legally in Korea you need a work visa. Apply for this visa from outside the country, preferably in your home country. Another alternative, if you are already in Korea on a tourist visa, is to apply for the work visa from a neighboring country like Japan or Hong Kong. Check with the nearby Korean consulate in your home country for the most up-to-date requirements for a work visa. Make sure you also gather your letters of sponsorship from the school that is sponsoring you for your visa application. You will receive an E-1 visa if you are teaching at an institution higher than junior college, an E-2 if you a conversational instructor or an E-5 if you are a working professional sponsored by a public relations firm or corporation.
Obtain a residence permit and a re-entry permit at the Korean immigration office within 90 days of your entry into South Korea.
Do your homework. Familiarize yourself with South Korean culture and customs, so you are respectful of their culture when you arrive. Western culture differs greatly from Korean culture and it is important especially as a teacher to know these differences.