1) Already have an ARC(Alien Registration Card)
You should notify the community service located near your new address within 14 days (unless you notify your moving-in within 14 days, you will be fined), following which you can get a stamp stating the definite date on your contract paper. Just visit a community service center with your contract paper and ARC to fill out the form.
2) Do not have an ARC yet
You do not need to bring a completed contract (this is done when you move in) when you go through the airport immigration. Just tell them the address of your place in Korea. You can complete the address registration for foreigners with the finished contract in hand after moving in. Bring your contract and passport to the Korean immigration office and they will do all the paperwork for you.
Once you are done with this process, your deposit is protected by the ‘HOUSING LEASE PROTECTION ACT’. This might not be familiar to you. If a building is put up for auction, you have the right to preferential payment that can be repaid, up to a maximum of 3,7000,000 won for buildings located in Seoul (the amount varies for different provinces and cities).