As this is my second time going I figure I might be of some help to those who are also thinking of living and working in Korea. So I am making a bit of a list of the things you will need in order to go over, most of these apply to Japan as well, but there are differences.
First off you will need a 4 year degree in any discipline, it really doesn't matter if you haven't studied language, English, or teaching. As long as you have a degree in something you are set.
Next you should work on getting your criminal background check (CBC). The first time I went to Korea you only needed a state level CBC, in 2011 the rules changed. You must obtained a Federal Level CBC, which means you will have to send a letter with your fingerprints to the FBI. I put this at step two because it can take up to 12 weeks to get this done and will be the longest step, so you should start early.
It would be a good idea to start looking for recruiters. There are hundreds, take your pick. The more famous ones are EPIK/GEPIK, grant it they are a bit harder to get into, they have a very elitist mentality. I went with HandS on Korea. A recruiter is useful because they will help find you a school with your preferences and you wont be charged a thing because the school pays for it, when you think about it, they are not looking for schools, they are looking for teachers. This is good and bad, on the down side most recruiters wont care all to much where they put you, they will get paid all the same.
A few other documents you will need to gather will be 3 passport pictures, a copy of your diploma, which will need to be notarized and then apostilled in your state, it does not have to be the state of your university, whatever state you are living in works. 2 letters of reference, your resume, a passport, and a Self health check, which is really nothing major, you print it off the internet and fill it out on your own.
While getting your documents, your recruiter is finding you a school, he/she will give you a list of schools and some info on each. you pick one and then you will have an interview over the phone with the school. But I wouldn't think of it as an interview, they just want to say hi more than anything. Now heres an important step I didn't do the first time, go online, there are websites that list schools in Korea that have been blacklisted along with reviews of the school. If you are going to a Public school (recommended) there is no need to look it up. But if you are going to a private school which I strongly strongly strongly discourage any sort of thought, then you are going to really want to look it up.
After you have your documents and sent them all to Korea, and you signed a contract with your school, you will receive a visa number. You will take that number with your passport and go to your nearest consulate. Near is a gross over statement, chances are your consulate is many states away. I think there are about 5 in America. I, like many, had to travel many states to get to my consulate in Chicago. Worst part of it was, you will only meet with him for about 5 minutes. You hand shake, say why you are going, handshake and walk out. It's a terrible formality one that should be discontinued or made easier.
Thats it, you will most likely meet the consulate a day or two before flying out, so all thats left is to pack your things, say your goodbyes and get ready to head out. Hope this helped in some way, and you are more then welcome to ask any and all questions.