In bold : those you nead in order to understand something in other dramas.
Those I didn't watch :
Reply - I read about it a lot on this forum but I can't bring myself to start one.
Winter Sonata - I actually never watched that one but I heard about it a lot in other dramas.
Modern time :
City Hunter - "Revenge heroe in modern time" genre, plus supposedly adapted from the manga of the same name (this was a huge scam even though the drama was great)
Coffee Prince - For gender-bender freaks (I'm one of them.), this is THE reference. Although I personally think this one is already old-fashioned.
Full House - I think this was my first kdrama, and it is the same for most people who started watching kdrama around 2004. When it aired, it was kind of the representative of 2 drama genres : "living under the same roof" and "in love with a celebrity". But now, it has grown a bit old-fashioned.
Goong - Just like Full House, this is a famous manwha adaptation.
Healer - Hidden identity genre with action.
I Hear Your Voice - Great plot, great music, noona romance and mind-reading
Kill Me, Heal Me - Romantic comedy about a psychological disease (DID).
You Who Came From the Stars - Romantic comedy with an alien!
Period dramas :
Bridal Mask - Revenge genre but during the japanese occupation which made it quite unique.
Hong Gil Dong - Just like Winter Sonata, you get some references to it in other dramas from time to time. This is a period drama about a Robin Hood type heroe.
Iljimae - Just like Hong Gil Dong.
Jumong - About the founder of Joseon
Queen Seon Deok - For many people, this is the best saeguk ever. A LOT of references to Mishil, the evil character, in other dramas.
Sungkyunkwan Scandal - a period drama AND a romantic comedy at the same time, that's the only one I know.
The Painter of the Wind - This beautiful period drama won numerous awards. Plus, you get to learn a lot about the importance of royal painters.