A hagwon is a sort of school where Korean students go for extra study in just about any subject. There are all-around academic hagwons, foreign language hagwons, computer hagwons, music hagwons, and even driving hagwons. A hagwon may be part of a chain or independent. It may have been around for a long time, or may be an uncertain new entrepreneurial adventure.
The hagwon at which I was employed is mainly a foreign language hagwon, but also has classes in other academic subjects. It has three English teachers who are from Korea, and three native English speaking instructors. It has been in operation for about five years, and was formerly (but is no longer) associated with the ECC chain. The students range from age 3 to their 50's, but most are later elementary school and middle school students. Most of the classes are an hour long: half an hour with a Korean teacher, and half an hour with a foreigner.
The basic work-week for one of the foreign teachers at my hagwon is 33 hours of class time, including Saturdays. Most classes are scheduled for between 2:00 and 10:00 pm. There are some holidays, but they may not be freely chosen - all days off are scheduled by the hagwon. The foreign teachers live in an apartment on the sixth floor of the building which contains the hagwon. They are hired on the basis of one-year contracts, which can be renewed. (A teacher who stays for a second year may be offered a more favorable contract).
Our contracts, which are basically standard, stipulate that the hagwon pay for the teacher's plane ticket to Korea. There is also a year-end bonus equal to one month's salary, which should cover the return ticket with some money to spare. Our salary is 1.5 million won (about $1300 US) per month, paid monthly. Early termination of the contract results in the loss of this bonus, and (in theory, at least) the teacher must also then refund to the hagwon the cost of his plane ticket to Korea. So quitting a hagwon job is pretty expensive. Housing (but not utilities) is provided free by the hagwon.
If you are looking for a hagwon job, anything less favorable than this contract probably ought to be rejected. You may be able to get better terms (like Saturdays off, and more frequent and more flexible vacations) if you ask your recruiter for them. But it's nearly impossible to tell what day-to-day life and work in your hagwon will be like before you get there. You should communicate with someone who is or recently has been working at the same hagwon before accepting a job, but even this is only part of the story. Unfortunately, you never really know until you've been there.