Gyeongbokgung's palace is bordered by the folk museum at the north end, and the palace museum at the south end. The National Palace Museum, unsurprisingly, contains a collection of artifacts used by the royal family at palaces.
Being full of goods only the elite ever got to use, it's impressive and beautiful, but the folk museum might be more interesting to me. There's no reason to skip either, though, because both are free and Gyeongbokgung Palace is certainly worth visiting.
Being located at the southern end of Gyeongbokgung, it's the first building you reach at the complex from Gyeongbokgung Station (Line 3) Exit 5.
The museum is opened weekdays 9:00-18:00 (closed Mondays) and weekends 9:00-19:00; admission is free. Right now, English tours are given at 15:00.