If you lose something: wallet, passport, phone

Things tend to come back in Korea. People leave a laptop on a café table while they use the bathroom. Still, if something is gone, there is an order to follow.
Lost on the subway: remember the station you got off at and roughly when. Items sit at the station office for a day or two, then move to the lost-property centre for that line.
Left in a taxi: your ride history has the car's plate number. If you paid by card, your card statement can find it too.
The police run a single national lost-property site, lost112.go.kr, where items handed in anywhere in the country are listed. There is an English page.
Lost passport: get a loss report from the nearest police station, then go to your embassy. A replacement will not proceed without that report.
For a phone, call your carrier and suspend the line first. Search for it second.