"Annyeong-haseyo" 안녕하세요
Greeting when people first meet
When people meet at the first time, Koreans usually say "Annyeong-haseyo"
This is the same words "hello" or "How are you " in America.
The literal meaning of "Annyeong-haseyo" is Are you well and at peace?
The origin of Korean greetings can be found in the Korean War. At the time people died of hunger or disease every day after waking up overnight. That's why people say hello to make sure they've been well all night long.
There is "Siksa hasyeot-eoyo?" with the same expression. This means literally "have you eaten" or "did you have a meal?", mostly elder people use this expression or we use these words around mealtime or right after.
So people usually respond "yes I have" even if they haven't. This greeting pattern is similar to how English speakers usually respond to the question "how are you" with " Fine" regardless of how they actually feel.
"Annyeonghi gaseyo"
안녕히 가세요
(Greeting when people go back. )
This greeting literally means "I hope you go back safely"
"Annyeong-haseyo" and "Annyeonghi gaseyo" are formal expressions in Korea. There are many different casual words as a greeting such as
Koreans are just trying to show that they care about you by asking the question.