I have decided to make this section available after I have received occasional inquiry from students in Korea so that I don't have to repeat myself answering for the similar questions, thus saving my and their time/effort. I hope this is useful to everyone. Although I work at RI, I am not really familiar with the admission procedure at RI any more because it has been quite a while since I graduated in 1992. The following answers are from my experience and the observations I have made for the last few years.
Not necessarily in order, but I think these are important in my opinion.
Excellent GPA, TOEFL and GRE scores (remember you are competing with
Chinese and Indian students whose TOEFL or GRE scores are really high),
Excellent recommendation letters from your professors,
Some publications,
Good match between your research interest and RI professors'
research interest,
Available research fund at RI or your research sponsor
in your research interest,
Personal contact (e-mail or phone) with a professor at RI and his
recommendation to the Admission committee on your behalf
Absolutely not if you are strong in the above factors. However, it is very competitive to get an admission to the PhD program at RI because RI provides scholarship to every student during his/her study with good monthly stipend. Typically RI admits 10-12 students every year. And it will take 5-6 years to graduate. Almost every year I see a few students from China or India, but no new Korean students since 1989. However, there was one Korean student admitted in 1999. He has received his Master's degree from MIT. But he decided not to come to CMU after all.
I believe most students have electrical/ mechanical engineering
or computer science majors. However, RI does not look for any specific majors
as requirement to the admission. Your research potential and research interest
are more important. I knew a student whose major was in philosophy and
worked in AI area for robotics application.
In general, a good engineering background
would do better as long as you can show your potential as a good Ph.D. student
by the forementioned factors.
I strongly recommend you to contact one or more professors at RI working
in research area which you are or will be interested in
and see how they respond to you. A professor must have
research fund to support for a new student.