As beginners we do not need to be intimidated by the Korean counting units because we also have counting units in English. For example a school of fish, a slice of pizza, a box of apple, a piece of paper. Here you can find a summary of the Korean counting units in a printable PDF for easy reference.
In fact there are no problems at all if you do not know any of these counting units because they can be omitted in a sentence. For example instead of saying 사과 한 개 (an apple), you can just say 사과 하나, which means the same thing. But using counters will make your meaning more precise e.g. is it one apple, one slice of apple or one box of apples. Similar to in English it is weird to say a wine but you’d say a bottle of wine.
This is the default counting unit for tangible objects in general.
사과 한 개 있어요.
There is an apple.
Counting unit for clothing e.g. pants, dresses, coats
드레스 두 벌을 샀어요.
I bought two dresses.
Counting unit for people
사람 세 명 왔어요.
Three persons have come.
Counting unit for animals
우리 집에서는 고양이를 두 마리 길러요.
Our family keeps two cats.
Counting unit for paper (piece)
종이 한 장
Counting unit for book
책 세 권
Counter for drinks (cup of)
한 잔 할래?
Shall we go for a drink?
Counter for drinks (bottle of)
맥주 한 병 주세요.
Please give me a bottle of beer.
Counting unit for machine e.g. cars, washing machine, computer
하늘에 비행기 두 대가 날아가고 있다.
There are two planes flying in the sky.
Counting unit for bowls of food
밥 한 그릇을 비우다.
He emptied one bowl of rice.
Counting unit for places
몸에 여러 군데 다쳤어요.
I hurt a few places on my body.
Counter for cases of incidents and accidents
김 형사는 이달에 사건 두 건을 해결했어요.
Police officer Kim solved two cases this month.