First of all, I recommend bringing at least $1000 USD of spending cash for the first month. Pack toiletries as if you were going on vacation, just so you have enough until you are fully settled into your apartment.
Between Costco, Gmarket, iHerb, and foreign marts, I’ve found that most things can be purchased here -especially if you’re willing to pay more. I believe in the old adage that if Koreans can live without it their whole lives, you can probably live without it for a year, but I’ll still humor your question with an educated response.
Bedding: I have never wasted precious luggage space this way, but friends swear by packing their favorite fleece blankets, jersey sheets, or fitted/flat sheets. All these items can be extremely difficult to find and/or expensive in Korea.
Birth control: If you prefer a certain brand of condoms or pills, stock up in your home country and bring them over.
Board games: Games to play with friends or with your students, I’ve brought cards (e.g. Uno) and board games (e.g. Apples to Apples).
Clothes: If you are bigger or taller than the average Korean, may want to bring a year of swimsuits, shoes, socks, underwear, bras, jeans, and/or sports gear (e.g. cleats and pads). Korea brags that it experiences all four seasons. Prepare for a bit of snow, a bit of monsoon, a bit of humidity, and a bit of sweltering heat. Ladies, lean towards being more conservative on your top half (wear cardigans with those tank tops), and gentlemen, you’ll look further out of place with baggy pants.
Comfort foods: I like to bring my favorite spices and hot sauces from home. I don’t know the laws of what is legal to bring into the country, but I have definitely had meat (i.e. Spam and beef jerky) confiscated from my luggage at the airport.
Electronics: If you really want to bring your favorite hair straightener or gaming console, read your cables. If they say anywhere on them “Input: 100-240v,” you can plug in your electronics with a little plug adapter. If you don’t have that cable, you should buy a power converter.
Toiletries: Deodorant is becoming more readily available, and womens’ make-up is plentiful —but many of your favorite brands may not be available in Korea. I, personally, bring over Tom’s apricot deodorant and Bare Minerals make-up, because I’ve never seen either in Korea.