Quintessential Korean Drama

I'm just getting into Korean dramas and really enjoyed Cinderella's Sister, despite (or perhaps because of) some makjang moments. Not sure if it's considered "essential", but it seems more mature than a regular rom-com. Interesting characters and a great performance by Moon Geun Young as Eun Jo.

The only other thing I've seen is Queen Seonduk and that one is an excellent historical drama and probably essential. Boys Over Flowers is considered "must see" and sounds like another good one to check out.

The dramas you need to watch, which I have also seen, are:
Descendants of the Sun
Healer
I Hear Your Voice
It's Okay That's Love.

The ones I know were super popular but I didn't really enjoy that much (and why):
Fated to Love You (too much comedy for my liking, I wished it was much more serious)
Gu Family Book (I wasn't too fond of it for several reasons including acting and computer graphics, but..)
Secret (the love story was too unbelievable imo; if you end up watching this, please take note that it's very heavy and people only either love it or hate it).

Dramas I know were super popular but I still haven't found the time to watch (as I am a relatively new and also quite busy drama-watcher):
Liar Game (there are other versions of it though, not only the korean one)
Empress Ki
Flower Boy Next Door (and other dramas in the Flower Boy series, they are all probably pretty cute and, well, not that mature)
Pinocchio

Seconded Cruel City, Liar Game, City Hunter, You Who Came From the Stars. ALL of them are must watch...

I like 'mature' dramas too, however, I'm not sure my definition of mature is the same as yours.
I like dramas that feature an older cast -- in their 30s rather than 20 something flower boys. While the story can be light-hearted, it shouldn't be WTF silly. The tone of the show should feel adult and there should be real, relatable adult themes that is handled in a way that doesn't defy common sense and logic.

I also appreciate when there's realistic intimacy (french kissing, love making) between characters who are supposed to be lovers.

With this in mind here are my recommendations:

Lovers, City Hall, A Gentleman's Dignity (all from the same writer) -- in order of my own preference. LOVED Lovers and City Hall. AGD is a bromance between 4 male friends and their respective love interests. There's a strong 'boys will be boys' vibe to it, so there are some silly, but not dumbed down moments among the friends. I personally thought it was a fairly realistic depiction of how men behave around each other. Outside of the friendship scenes, I thought the romantic story arcs were handled maturely and reasonably well done.
 
Can We Get Married, Can We Love (both from the same writer) - I like the sequel (CWL) better than the prequel (CWGM), but both dramas were less than spectacular. Still they're very quintessentially Korean and worth a watch.

Coffee Prince - this is a classic and very popular among viewers. It's a nice romantic drama with a pair of great leads, however, the story got really draggy in some parts. Nothing the FF button can't fix though, so still think you should give it a go.

If you want to try some melodramas, I found these two pretty decent:

Padam Padam

Que Sera Sera

If you're looking for excellent character development, intelligent story, but with not much romance

Golden Empire

is an excellent choice. I love the writer of this drama, because while his dramas deal with quintessential Korean themes, the quality of writing is often far above the normal K-drama. More on par with great American TV shows like House of Cards and Homeland (in terms of writing, not story).

Cruel City is also okay to watch, if you want a bit more excitement in your drama, but of less quality in writing and story than GE.

I liked Nine: Nine Times Time Travel as well, a good mix of romance, time travel and suspense.