As an amateur / new watcher of K-dramas, I’ve needed to learn some Korean concepts to better understand the plots.
If you are new to Kdrama here’s some things that you might find useful to know:

Warning: I am by no means an expert on South Korea in any way shape or form.
Names
Korean names are given as <surname> <first name>. Sometimes this can be confusing.
When Korean names are translated to a name using the English alphabet the name may be translated differently by different translators. A good example is a character has the surname “Ri” in the subtitles but their name is shown on an item in the film, say an envelope, as “Lee”.
Using just the first name is only for immediate family or super close friends.
Korean people can use different names to their Korean name for English or Japanese contexts.
I often lookup a wiki on the show, and have it to hand when I get confused about names and who is related to who.

Lost In Translation
Speech is very polite – there are 7 levels of politeness. This isn’t translatable into English, so sometimes the subtitles will say “why are you talking informally?”, which appears to be out of the blue, but is actually a result of these levels not translating to English.

It seems to be fairly common in south East Asian shows to have text appear explaining certain concepts, or commenting on events. This is why you will sometimes see Korean text on the screen.

Chaebol
A chaebol is a super rich family; essentially kingdoms but they are corporations.
The Chairman is like a king, and the President runs the corporation. Often the Chairman is also the President.
A child of the Chairman can be nominated as the successor to the Chairman (normally this is the eldest son, who is like a prince). This appears like incredible nepotism, but is the equivalent of Murdoch giving his kids jobs. Cue drama of children fighting over who is successor.

Arranged marriages are common between chaebol children and are similar to a company merger. Cue drama of son wanting to marry a poor woman.
Illegitimate children of chairman are often adopted into the family, and it is not scandalous for this to occur. Cue drama about “real” children vs “adopted” children.
Year of Birth Matters
Korean people know or guess the age of people they are talking to, and speak and act accordingly.
Older people (hyeung, unni, oona) are meant to look after younger people. In exchange the eldest / most senior is (meant to be) respected. Cue elders chiding younger people for speaking informally to them.
If you are the youngest in a group (maknae) everyone else will be your hyeung or unni.

Some translators leave in the “hyeung”s in the subtitles. Some use terms like “brother” but if you listen out you can hear the person say “hyeung” or whatever.
Eating and Drinking
South Korea still has communist-ish culture. Eating with colleagues (either lunch or dinner) is more frequent and more expected.
Drinking a lot of alcohol (often soju) is not unusual. Getting drunk and blacking out isn’t shameful, even with work colleagues.

Celebrating relationship milestones, like first 100 days, is common. Cake is expected for celebrations.
Skinship
Skinship = Korean men hold hands, hug each other, share beds. No big deal.

Touching someone of the opposite sex is a huge deal. Cue lots of close ups of someone gripping someone’s arm, which is a.. Big. Deal.
History
You don’t need to be an expert in Korean history to watch historical series. It would help to be aware that Japan, Russia, China and the USA were political players in Korea for many years.
Technically North and South Korea are still at war.
How To Watch
The fast forward button is your friend.
Lots of endless shots of people looking at each other or repeated views of the same action from multiple points of view can be skipped.

The end of the previous episode is repeated at the start of the next episode.
Make sure to watch to the very end of each episode, as often after the recap images, there is a short little section that gives a little backstory or subplot event.
K-Drama Bingo
I love this list of tropes and cliches.
