Post office boxes are very common throughout cities and towns. You’re bound to see a few just walking down the street outside your house, usually on the corners. For regular mail going somewhere in Korea, you stick on a stamp (that you buy from the post office), and throw it in a red mail box.
But most of us will probably not be sending letters within Korea, but International mail. And for that you need to visit the post office, which is more of a hassle, and my experience is that post offices are not as common as in my home country.
There’s bound to be a post office somewhere close to your house and the best way to find out is to ask around. Co-teachers or colleagues will be the best help. Non-standard size mail or packages also need to be sent from the Post office, and for a parcel you need to fill in a customs declaration form.
You can write addresses in English, this almost never poses a problem. But, as always you have to make sure the post code is correct. So when you get hold of a Korean address (or your parents want to send you a package from abroad), make sure you include the post code in the address. Post offices are open 9:00am-6:00pm (until 5:00pm from Nov. – Feb.) Post offices are closed on Saturdays, Sundays and holidays.