Like with all international travel, you will need a valid passport to study in Korea. The consulate officially requires that your passport be valid for six months AFTER your stay is completed. Also, one blank visa page for entry/exit stamps is required.
Visas are not required for stays of shorter than 90 days. If you are staying for an entire semester, here is a list of things you will need to apply for a visa:
• Passport: don’t forget to make a copy and send the original
• Visa application form: download here 
• One recent 2” x 2” color photo: straight on headshot from shoulders up on a light background
• Official invitation: the acceptance letter from the university. This must include a letter detailing the reason for your stay, the duration of your stay, an address, and contact information. All of this should be provided in the acceptance packet. Make copies of these forms and send the originals
• $45 money order made payable to Korean Consulate General
• Self-addressed pre-paid envelope for return mailing
Visas normally take about four working days to process. Expect anything from five days to two weeks. There is a rush visa option available if you have specific time constraints.
If your residing address is in the Midwest, chances are you will have to send your visa packet to the Korea Consulate General in Chicago, Illinois. Visit their website  for more up-to-date visa information.
As with all countries, the requirements for entering South Korea may change overnight. For example, the H1N1 outbreak has altered the procedures and increased the paperwork for many countries in Asia. Make sure to contact your study abroad advisor and keep abreast of current news that might affect the region to avoid getting trapped in an airport or sent home.