I spent my first night in Korea at Seoul Backpacker’s Hostel which was really great value for the price. It is located only 15 minutes walk from Insadong and only five from one of the royal palaces. It is easily accessible from the airport, and their directions, both online and available in pamphlet form from the Tourist Information booth at the airport are great, and it is only 10 minutes walk from the Anguk-dong bus stop on Airport bus 602-1. The hostel is just around the corner from the Japanese cultural center, which since I got here has been defended by at least two dozen Korean riot police with really big sticks and riot shields.
Single rooms cost only 27,000 won when I stayed there, and you can stay in a “dormitory” style room for only 17,000. They have a whole range of free services like a kitchen, toast for breakfast, free internet, up to 2 weeks storage for baggage after you check out, a refund on your airport bus if you stay more than five days, and various discounts on tours and cultural activities. The rooms are nothing fancy, youth hostel quality, but have a private bathroom and my room had a fridge and TV. The staff are super friendly, helpful, and speak English and apparently Japanese as well. They also have half a dozen beautiful dogs they keep around.