Fill out the DS-11 passport application using the “Passport Wizard” (https://pptform.state.gov/) and submit your old passport, proof of identity, passport photo, and fee ($135 for adults, $105 for children). You can get a new passport in 2 to 3 weeks. If you submit your old passport, then you do not need to come to the Embassy. However, if you cannot submit your old passport (because is it lost or stolen), then you will need to make a personal appearance at the Embassy in order to get a new passport.
Frequently asked questions about passports are answered at: https://kr.usembassy.gov/u-s-citizen-services/local-resources-of-u-s-citizens/frequently-asked-questions/