Gwanghwamun Gate (광화문) is in the top of the 'must see list' in Seoul if you are interested in the South Korean culture in history.
It is the main gate to the Gyeongbokgung Palace (경복궁) and one of The Eight Gates of Seoul which were part of the Fortress Wall of Seoul and surrounded the city in the Joseon Dynasty (only six of them exists today).
The sad fact is that this gate is not the original one as it has been destroyed during the Japanese occupation of Korea and rebuild in 1968.
Seeing the Gwanghwamun Gate is a great experience on its own but if you want to spice it up a little bit visit during The Royal Guard-Changing Ceremony.
Sumunjang (Royal Guard) Changing Ceremony
10:00, 14:00 / 20 minutes per ceremony
Gwanghwamun Gate Guard-on-Duty Performance
11:00, 13:00 / 10 minutes per ceremony
Gyeongbokgung Station (Orange line, line 3). Exit 5 is the closest exit.
I recommend to take the line 5 (Purple), get off at the Gwanghwamun station and use exit 9. You will end up at the huge Gwanghwamun square from where you can walk around the statue of King Sejong (who instituted Hangul, the Korean alphabet) and come to the gate from the front.
Where to go from here
Gyeongbokgung Palace (경복궁) or Bukchon Hanok Village and Insa-dong.