There are quite a few popular tourist attractions in South Korea! I'll also include places that might interest you, not necessarily tourist attractions...
Sightseeing:
If you'd like to experience historical Korea, you must visit one of the many cultural villages! It really shows you Korean tradition and culture and you really feel like you're a part of the old villages. Here is a list; you can google these:
Andong Folk Village
Andong Hahoe Village
Cheonghakdong Village
Jeonju Hanok Village
Korean Folk Village
Naganeupseong Folk Village
Namsangol Hanok Village
Korea has some really beautiful scenic places if you're interested in looking at natural attractions. The most famous would be Jeju Island. If you get a chance, GO! It is truly breathtaking! But, here are others just in case:
Chaeseokgang River
Geoje Haegeumgang
Goseokjeong Pavilion
Jungmun Daepo Coast Jusangjeolli Cliff
Nari Basin
Sangumburi Crater
Seongsan Ilchulbong Peak
Seopjikoji
Mt. Jirisan
Yongduam Rock
If you want to see museums and whatnot, here are a couple. These are definitely not all of them..just a few. It really depends on what you're interested in:
National Museum of Korea
National folk Museum
Kimchi Field Museum
Abraham Park Kenneth Vine Collection Museum
Andong Folk Museum
Andong Soju Museum
Buyeo National Museum
Cheongju National Museum
Hanbok Museum
Icheon World Ceramic Center
the list goes on and on...
City Life/Other:
Myeong-dong: This place is a MUST! It's basically a massive shopping center. It also has great places to eat. An awesome place to go. It's practically always crowded, even at night. If you're a big city place, you'll love it here!
Daehakno (University Street): place for younger people. nice little place to go. Center for performance art.
DMZ: This is the dividing line between North and South Korea. Honestly, I didn't find it That interesting, but if you have time, go. It's pretty cool
Everland: This is an amusement park. In terms of size and rides, it doesn't even come close to Disneyland. However, it's a fun little place and you'll enjoy, especially if you have little ones along.
Seoul Land: another amusement park. In my opinion, not as good as Everland
Coex Aquarium: not all that interesting to tell you the truth. I think you'd find better ones in the U.S.
Insadong: This place sells lots of traditional Korean things, center for art, lots of good restaurants. Really get a feel for old Korea. You should go.
Seoul: Really, if your a city person, this place is it. I didn't list many specifics but there's a ton of things to do here. Great architecture, great atmosphere, wonderful food, very upbeat. I don't know what age you are so I can't really recommend specifically...
phew..I really hope this helped. If you have any questions, I'd be happy to answer. I would have had better suggestions if I knew your interests and age. Anyway, good luck!