The Korean universities have excellent research facilities and have several government grants. But the Korean professors are extremely choosy to select the right candidate for their laboratory. The Koreans in general like to work hard and produce good quality outputs with good publications.
There are several governament scholarships which a student can apply through the Korean Professor:
1. KRF, Korean Research fellowship for foreigners
2. Brain Korea Fellowship
3. Sometimes individual universities support few students
4. Sometimes professor has several projects by which he can support you
The tution fees for the university is generally high, which can be sometimes 50% wiaved based on your scores.
All Korean University has an International Department which takes care of all the foreigner students.
Its better to live in the university dormatories which are cheap, otherwise if you cannot eat the Korean food, it better to take a studio apartment with a kitchen, near the university.
If you are interested in Ph.D, you should choose your professor carefully, with the same reseach background.
Generally, Korean university provides you with free korean language courses conducted by some students and also the university gives a six months korean langauage course which is covers the basics.
The few of the universities of Korea:
1. KAIST
http://www.kaist.edu/edu.html
2.POSTETCH
http://www.postech.ac.kr/
3. Seoul National University
http://www.useoul.edu/
4. Korea University
http://www.korea.edu/
5. Hanyang University
http://www.hanyang.ac.kr/english/
6. Pusan National University
http://english.pusan.ac.kr/html/00_main/
Good luck for the aspiring candidates!