You'll get different answers based on who you ask, but I've compiled what I believe are the essentials to living in the Land of the Morning Calm.
(Please keep in mind that this is from a female's perspective on living in Seoul. If you're a guy, you'll probably need less. If you are moving to a smaller city or the countryside, you may want to pack more depending on how fast you can take a trip to Seoul.)
The first thing to do when packing to move abroad is to check the baggage allowance of the airline that you're traveling with. This website seems to have a pretty comprehensive list of all the airlines out there and how much they let you bring on the plane. Once you have figured out the allowance that you're working with, decide on what you want to bring.
Korea has four distinct seasons, so it's appropriate to pack accordingly. There are numerous shopping areas in Seoul and other large cities of Korea. While Asians are typically smaller than Westerners, there are many stores that cater to larger sizes. Forever 21, H & M, and Zara are a few retailers in Myeondong that offer Western sized clothes at affordable prices. They carry men's apparel, as well. Additionally, SPAO and Uniqlo are two Asian brands that carry some larger sizes as well. As an added bonus, Uniqlo offers free tailoring on jean purchases.
Despite the fact that western sizes are available, there are a few essentials that I believe are worth bringing from home: a bathing suit, a couple pairs of blue jeans, a good winter coat, and if your employer requires it, some nice dress clothes.
For girls, I'd recommend taking lots of bras and panties. Even if you are smaller framed, the size conversions can be a bit complicated and most of the stores that I mentioned will not allow you to try on undergarments before purchasing them. Finally, shoes. I have unusually large feet for a girl (USA size 9.5) so it's almost impossible to find shoes that fit me in Korea. Forever 21 and H & M will occasionally have larger sizes (8-10) in stock but they get snatched up pretty quickly. So, bring along shoes for all occasions. Don't forget your socks, too.
Although times are changing and Korean girls are beginning to wear less, be mindful of the fact that it's still considered inappropriate for women to show off too much shoulder or collarbone here in conservative Korea. Don't even think about walking around with your cleavage showing. If you do, expect to have a hunched over grandmaw give you the evil eye all the while yanking your shirt up. I've seen it happen. Leave the revealing clothes at home, as you won't wear them when you get to Korea.
After gathering your clothes, it's a good idea to use Space Bags to create as much room as possible in your luggage. They're also great for storing off-season apparel when you get settled into your place in Korea.
Non-clothing items that I'd suggest one bring to Korea are linens and towels. Fitted sheets are no where to be found in Korea and even top sheets are quite pricey ($40USD and up). If your school is providing your accommodations, they will most likely give you a blanket and a mattress topper and even if they don't, they are easy enough to find upon arrival. If you're unsure of the size of your bed, ask your school. If they are unable to tell you, go for Queen size. Depending on the quality, linens can make your luggage a bit heavy but trust me... the comfort will be worth it. As far as towels go, most Koreans use hand towels to dry off after bathing. If you'd prefer something a bit bigger, bring them from home. Pack one normal towel and one beach towel in the case you head out to Jeju or Busan on vacation.
Western brand toiletries are now easier to find in big cities. Pantene, Dove, Head and Shoulders, Listerine, Neutrogena, Aveno, and even some luxury brands are quite common at stores like Olive Young and Watson's. Unless you're super picky about the brand you use, don't worry about packing shampoo, face wash, conditioner, body lotion, shaving cream, body wash, or mouthwash, as they are too heavy. Of course, travel sizes are okay to get you through the first few days of your stay. I do recommend bringing your own deodorant and toothpaste, as Korean brands are quite different than our Western ones. Sunscreen is something that is smart to bring from home, as I've heard the SPFs in Korean brands aren't as closely regulated as they are in the West.
Depending on your skin tone, makeup can be difficult. I'm very fair and didn't have much of a problem finding foundation to match my color, darker tones can be hard to come by.. Brands like MAC and Clinique carry shades that favor Asian skin tones so it's good to bring what you use from home. You can find cosmetic shops on every block in Seoul. Many, like Missah or Innisfree, carry decent quality eye shadows, liners, etc. and allow you to try them before purchasing.
Tampons, while available, are still few and far between. They're almost twice the price of what they are in the States so if you have extra room, throw in a couple months' worth. These, too, can be found at Olive Young, Emart, and Watson's which are located throughout the city.
Pharmacies are often easy to find in Korea and many pharmacists in Seoul can speak some English. However, I think it's a good idea to bring the basics: immodium, Pepto, Tylenol, cold/flu medicine, antibiotic cream, and anything else you may commonly use. Your doctor at home will most likely write you a script for a prophylactic antibiotic in case you may need it in Korea. Many doctors in Korea speak English, especially at international clinics, so don't fret about this one.
It's also essential to bring along copies of all your important documents in your carry-on luggage. I always keep copies of my passport, driver's license, visa, birth certificate, and social security card in the case I lose the actual documents. Remember to keep them in a place separate from the originals.
I've had a laptop since I've been in Korea and it has been my link to the outside world and my home. Whether catching up with friends on Facebook, calling parents on Skype, or watching movies, I use it daily. I also purchased an iPod Touch which has kept me occupied on cross-country trips or on the subway. Some sort of hand-held media device will come in handy. Also, as you'll be seeing lots of new things, you'll want to capture your memories with a camera. It's inexcusable not to bring one.
Finally, if you still have room in your luggage, bring along a few small things that remind you of home. Photos of family and friends, a nice candle, a cute picture frame, or a meaningful trinket will go a long way in making your new home comfortable and cozy. Your students and new friends will also enjoy seeing pictures of your life across the world.
As you are packing, don't forget to periodically check the weight of your luggage. You may want to invest in a luggage scale to ensure the weight is accurate, or do it the cheap way by holding your suitcase while weighing yourself on the bathroom scale. Don't let it stress you out too much. In the case you run out of room, the US Postal Service offers a somewhat affordable international shipping service. Your friends or family at home can ship whatever you need in a decent sized box for a flat rate. I'm sure other countries have similar services.
Don't forget to label your luggage properly with good quality luggage tags. Double check to make sure the person checking you in at the airport applies the correct destination tag to your bags. This will ensure that your baggage will arrive when you do, leaving you less-stressed and more optimistic about your new home.
So, all that's left to do is get on the plane and GO! Good luck on your new adventure.