Besides the most important areas listed in Key Areas (Part 1), there are a couple more key areas of Seoul worth noting if visiting more than a couple of days.
Yeouido is an island on the Han River, roughly south of Hongdae. It's the home of many banks, corporations, and government buildings alike, but is perhaps most famous for its Han River Park.
When the weather is nice, the park is full of endless rows of picnic blankets, and people enjoying snacks, like fried chicken and beer (치맥). In fact, since you can get food delivered to almost anywhere in Korea, people often get food delivered directly to the park. It is also allowed to drink alcohol in most public parks in Korea (although this is a slightly controversial political topic), including Yeouido Han River Park.
Just west of Gangnam is the Jamsil area. Just like Gangnam, this area is relatively new in Seoul's history, and is most famous for the Lotte World Tower -- the tallest skyscraper in Korea, and the sixth-tallest building in the world.
At the bottom of the Lotte World Tower is a large shopping mall, the famous Lotte World theme park, as well as a beautifully laid-out lake and park area. The lake is called Seokchon Lake, and it is probably the most famous place for viewing cherry blossoms in Seoul during the cherry blossom season.
Similar to Hongdae, Seongsu is a university area, known for endless aesthetic, cute, and hip places ranging from cafes to food to nightlife. However, Seongsu has a less strict stereotype of being for "young" people than Hongdae, and feels much less international. There are far fewer foreigners visiting Seongsu, and it's much less common to hear English.
Lastly, the Dongdaemun area is mostly famous for having the Dongdaemun Design Plaza (DDP). It is architecturally a very nice building to visit, but unfortunately there is not much to see or explore inside (unless of course you are part of a fashion event).
Notably missing from this list is the area around Seoul Station. One might expect that taking the express train from the airport and arriving at Seoul Station, you would be welcomed with a beautiful and modern area. While there is plenty around (there are some people that like this area), it is not the nicest part of Seoul.
Lastly, it should be mentioned that this overview has been by no means exhaustive. No matter how much of the city you've seen, there is somehow always a new area to discover. One might randomly stumble on an old industrial area recently becoming a hip zone for bars and restaurants, and often only known by people living nearby (take 문래 for one example). Or you might discover the best Thai food in Seoul is randomly hidden far away in the outmost corner of Guro. The world remains inexhaustible, and Seoul is no exception.
