One of my favourite places in Seoul is Insadong (인사동) and Ssamziegil (쌈지길), which is part of Insadong.
This part of Seoul can be quite busy at times because it's a well-known tourist location. However, you definitely have to drop by before leaving Seoul.
This place is so rich in culture where the old and the new of Korea co-exist. It's one of those places that makes you stop and think about Korea, not only what it is in the 21st century, but also about the long historical past that this country has.
If you initially became interested in Korea through Hallyu (K-pop or dramas), or you are more familiar with Korea as an economic miracle of the 20th century, then this will show you another part of this country.