EPIK (English Programme in Korea)
EPIK is probably the most popular way to teach English in Korea and most of the people you meet (at least where I live) are on the EPIK programme. It is run by the Korean government and if you get a job with EPIK you will be placed anywhere in Korea at either a public elementary school, middle school or high school, you will work from around 8.40 to 4.40 Monday to Friday and you will have 22 hours of class time within that schedule. The pay varies from 1.8million won if you have only got a bachelors degree to 2.8million won with experience/a masters in Education. You also receive housing or a housing allowance of 400,000won, 18 days holiday, a signing on bonus, a renewal bonus AND free return flight airfare. For the full range of benefits check out their website. Everyone (should) also have a co-teacher who will do a great job of keeping the students in check and will help you out in the classroom and with things such as lesson planning and your curriculum.
While this all sounds like a pretty good deal(it is!) there are a couple of things to watch out for. Firstly unless you know someone in the office of education at the city you want to teach in you can not choose the area you will be living in. This means you could be placed in a massive city, a provincial town or (like one of my friends) an island with a population of a couple of hundred and no other English speakers. Obviously this last scenario is very unlikely , but it is still worth thinking about how you would deal with being placed in various locations.
The second thing is that as most people are placed in housing, you can end up either pretty far from your school or pretty far from your friends. A significant number of people where I live have commutes of over an hour, the positive of this is you are close to the city and your friends.
Hagwon's(or private academies) basic package is very similar to that of EPIK. The benefits that a hagwon will offer that EPIK doesn't is that you have complete control over where you will live, as you apply directly to the hagwon. There are usually a range of different hagwons with different working hours available, so if you not a morning person, get a job that starts in the afternoon! Like everything the hours vary between hagwons, with most offering you an average of about 8 hours either 9-5,12-8 or 2-10. However I have seen hagwon's offering a working day of only 2.30-8.30! Add this to the fact that your accommodation will almost certainly be close to both the city and your hagwon and your EPIK friends who leave for work at 7.30 at don't get home till 6 will be very jealous! On the flip side I know people who work split shifts that start at 7am and don't fully finish until 9pm. Check your contract!
The negatives with working in a hagwon is that when your at work you will always be teaching, there is very little desk warming time, so even if you do get a job with a 6 hour a day contract you will still be teaching much more than EPIK teachers. You also have no co-teacher which is usually ok as the classes are smaller, but it does mean you have to be able control a class on your own. The worst thing for me about Hagwon's is the holidays. You will usually get around 10 days(although this can be less) and you will have to take them when your hagwon boss wants you to take them. Some of my friends have had their 10 days split into three meaning they have never had a full week off in two years of being in Korea.
The final(and in my experience unlikely) problem with hagwons is that there are stories of hagwons closing leaving teachers without a job, and of them trying to get out of paying the final months pay and airfare. However these are very rare situations and as long as you can speak to some of the current staff you should be ok. To look for hagwon jobs and see for yourself the kinds of conditions available check out these websites.
TaLK (Teach and Learn in Korea)
The final option I will discuss is the TaLK programme. Like EPIK it is run by the Korean Government. With this programme you will be teaching after school classes at a rural elementary school. Your commute will either be long or you will live in a rural area, the pay is less than EPIK (1.5million won, more than enough to live on, save money and have a holiday) and you have a month long unpaid vaction BUT.. you only work THREE HOURS a day. This is great for those of you that want to spend time learning Korean or focusing on a hobby. Talk also provides 6 month contracts so it is great if you are not sure about teaching or coming to live in korea for a longer time.