Well, you have touched on one very large and complicated topic for some.  Anything dealing with government bureaucracy, regardless of the country, is this way.  

In a nutshell, E-2 visa holders should visit the local immigration office in their area of residence in person to complete this process. Permission to extend your working visa must be acquired before expiry of current permission of stay.  You can request short-term extensions up to 30 days if you plan on leaving the country.  If you wan to extend for a longer period (i.e. 1 year), Korean Immigration states that you must apply 2 months before the date of expiration of your current E2 visa. Otherwise, you will pay the penalty according to Article 25 of the Korean Immigration Act.

Remember the following is/was necessary when initially applying for your E2 visa. It may not be necessary for an extension but best to check locally because you don't want to have to reapply/resubmit for another appointment if they ask for these:

1. An FBI criminal background check.  See http://www.fbi.gov/about-us/cjis/background-checks

2. 2 notarized and apostille copies of your degree (can be done at the US Embassy in Korea).

3. A medical check done within South Korea that tests for drugs and HIV.

South Korea actually has an e-Application center online at http://www.hikorea.go.kr/pt/index.html.