There are three types of counters in Incheon (and most other entry points in Korea, like Gimpo airport, Jeju airport, etc):
- Manned counters for Koreans.
- Manned counters for foreigners (residents and visitors).
- Electronic gates for Koreans, foreign residents and registered visitors.
For the latter, a few countries and regions have signed a reciprocal agreement with South Korea. The US, Germany, Hong Kong, and a couple of others I think. You need to register with Korean Immigration, online, pass once through a manned counter, and then register physically on departure. After which you can use the electronic gates.
Korea doesn't stamp passports anymore. At manned counters for foreigners, on entry, the Immigration officer gives you a small piece of paper recording your personal details, with date of entry, and date by which you have to leave.