Apartment Number
The apartment designation is almost always 호. If you live at Apartment 100, this is written as 100호.
Building Number and Street
A way to independently find out your mailing address, is to find the address plaque on your building. It is usually by the front door. The plaque will look like one of these:
A comma is added between the building number and street number. The correct romanized version of these buildings would be "209, Sejong-daero" and "35, Jungang-ro". It doesn't seem to matter that much. Letters addressed to either "35 Jungang-ro" or "35, Jungang-ro" arrive just fine.
Zip Code
After finding out your street name, go to http://www.juso.go.kr/openEngPage.do and type in the street address. You can type it in English or Korean. From there, you can get your six-digit zip code.
When typing in English, note that the formatting occasionally gets mixed up. If "Sejong-daero" doesn't work, try adding a space like this: "Sejong -daero".
Address Formatting for domestic mail (Korean)
The Korean format for writing addresses is reversed, compared to the Western format. Korean addresses go from bigger to smaller, which is the reverse of the smaller-to-bigger format in most English-speaking countries. The Korean system uses "South Korea, Incheon, Namdong-gu", instead of the English-language format of Namdong-gu, Incheon, South Korea.
For example, this is the Korean address for Bupyeong Post Office:
인천 부평구 길주로 415
(Incheon, Bupyeong-gu, Gilju-ro 415)
Address Formatting for international mail (English)
To write the address for Bupyeong Post Office, in the international format is:
415, Gilju-ro, Bupyeong-gu, Incheon, South Korea
2nd Example
For the second example, let's use the Sangok-2-dong Post Office in Bupyeong-gu. Their Korean address:
인천 부평구 마장로272번길 95
(Incheon, Bupyeong-gu, Majang-ro 272beon-gil, 95)
Now, when we change it to international formatting:
95, Majang-ro 272beon-gil, Bupyeong-gu, Incheon, South Korea