휴일 means any kind of holiday for someone. It includes private holiday, public holiday, and Sunday.
휴일에 가족과 뭘 하실 계획이세요?
What will your family do for the holiday?
오늘은 휴일이어서 나는 하루 종일 집에 있었어.
Today was a holiday and I stayed at home all day long.
공휴일 means public(national) holiday.
1975년 이래로, 어린이날은 공휴일이 되었다.
Since 1975, Children’s Day has been an official holiday
우리는 1월 1일을 공휴일로 삼고 있다.
We keep January the first as a holiday.
연휴 means consecutive holidays of two or more days.
이번 주엔 연휴가 3일 있어요.
We have a 3 day-weekend this week.
그 식당은 설날 연휴 동안 문을 닫았다.
The restaurant was closed over the New Year holidays.