Sssooooo much history to cover here. You could spend a month in Seoul alone.

Nav:

Gyeongbokgung Palace: Official palace of the Joseon Dynasty. Originally built in 1395, the palace was burned by the Japanese during a regional campaign. What you see today is a recreation.

Changdeokgung Palace: This UNESCO World Heritage Site is a 'modern' 15th century palace build to augment Gyeongbokgung. Replaced palaces became residences for former kings and princes.

Demilitarized Zone (DMZ) and Imjingak Peace Park: Tours are available of the DMZ with approval from the South Korean military. You can't walk up to the fence. A few of these photos are though a telescope lens at a museum / visitor center near the border. The North and South Koreans would try to put up a taller flag post than the other side. North Korea would play propaganda on loud speakers until the South Koreans played Gangnam Style and they were offered a truce (on music).

War Memorial of Korea: Every war in Korea is a Korean War, not just the 1950s war with America, Korea, and China. The respect South Koreans have for the United States and United Nations forces is real.

Nosh:

Gwangjang Market: A large indoor market with lots of food options.

Namdaemun Market: A large outdoor / indoor market with food and merchandise.

Other Korean Specialties:

Nip:

Kiwa Taproom: Craft beer with a traditional Korean atmosphere.

Terra and Soju: Terra is a national brew available at many restaurants. Soju is like sake, a rice wine. Cheap and packs a punch. Can come in plain or flavored varieties.