This is one of the most popular forms of work for expatriates and other foreigners with the travel bug, as it pays quite well and there are plenty of jobs available for anyone with a four-year English degree. You don't even need to speak the language (though it will, of course, help you get a job).
Koreans are extremely proud to the point of being touchy about their ethnic heritage. There is an institutionalized practice that different ethnic groups are ranked according to quality, intelligence, etc., with lighter-skinned individuals at the top of the heap and darker-skinned individuals at the bottom. This makes it difficult for dark skinned travellers to find English teaching jobs.
There are generally two places you can teach English in South Korea, and each has its benefits and drawbacks. You can teach English in public schools, or you can teach English in private academies, known as hagwon.
Public schools pay less, but they're generally always going to provide solid employment. Additionally, they are more accommodating toward "difference". Meanwhile, hagwon tend to pay very comfortable salaries, but because they're privately run you have to make sure you do your due diligence. Some directors are unscrupulous or just plain incompetent, while others may treat you poorly because you're a foreigner.
Additionally, you may give tutoring lessons privately, but you should make sure to have a work visa in order to do so or else you risk fines and deportation. Generally, tutoring doesn't get enough money to get by on its own, so the prospective English teacher should search for a job.
With one of the best English teaching systems in East Asia, South Korea is a beacon for many Western students and college graduates looking to see more of the world through travel while supporting themselves along the way. 99.9% of the time, there will be no incidents of bias. That number rises to 99.99999% as long as you remain positive and open-minded.
It is a beautiful country with an array of lucrative possibilities, and it would be a shame if you didn't take them out of fear. There are enough jobs available that if one job doesn't work out for you, you will be able to find another one within a few days.