Insadong is a shopping district near central Seoul and consists of a 1/2 mile long main street with countless alleys branching off of it. This is a great place to find things such as traditional touristy souvenirs, unique artwork, and beautiful pottery. It is closed to traffic on weekends, providing space for street vendors, parades, and entertainers. Touring Gyeongbokgung Palace and then shopping in Insadong makes for a great Seoul day trip since they are near to each other.
One street over from Insadong is a street nicknamed "monk alley" due to the large number of stores selling buddhist monk clothing, shoes, accessories, and religious items. There is also a large temple called the Jogyesa Temple. It is a beautiful temple with 3 large seated gold buddhas inside. It's worth a look if you're in the area.
Dongdaemun is... excess. For example, if you are looking for a pair of shoes, there are 10,000 pairs to choose from. If you want a scarf, you could spend all day looking through all that are available. This market is mostly indoors in large, multistory buildings like the one picture below. This huge marketplace is pretty well organized according to item. You just need to find the building and floor that has what you're looking for.
Namdaemun is one of my favorite shopping districts because it is a large, primarily outdoor, traditional marketplace. It's a great place to bargain for deals, try some street vendor snacks, take in the atmosphere, and for general people-watching.
"Shopping is a woman thing. It's a contact sport like football.
Women enjoy the scrimmage, the noisy crowds, the danger
of being trampled to death, and the ecstasy of the purchase."