Near the center of Seoul at the top of Namsan Mountain sits a tower, the North Seoul Tower (CJ Tower, officialy). From the top of the mountain, and the tower, the panoramic views of Seoul are vast and impressive.
A trip up the mountain is well worth your time, reservations can be made to eat in the revolving restaurant at the top of the tower while below martial arts performances are carried out amongst old women spinning cotton candy. In the midst of all these spectacular things I find the sculptures of Parisian artist Cedric Le Borgne to be the best surprise. The method in which he portrays the human form floating through air is simply whimsical.
To get to the top of Namsan you have three choices of transportation: hiking (which takes the longest but is free), bus (reasonably priced but poor views) and cable car (a little expensive but offers greats views).
Shuttle buses, sometimes free, run from Namdaemun market just North of the mountain is an ideal spot to find Seoul's yellow colored buses, these buses are known for stopping in popular destinations.