休日 and 休暇: days off and leave
Distinguish days when work or school is off from a period of leave.
Referring to a day off
休日 is read きゅうじつ and means a day off from work, school, or similar duties. 「休日は家で過ごします」 describes the usual practice of spending days off at home.
「次の休日は水曜日です」 says that the next day off is Wednesday. A 休日 is not necessarily a weekend day or public holiday.
Taking a period of leave
休暇 is read きゅうか and refers to leave from work or studies. 「来週、三日間の休暇を取ります」 says that the speaker will take three days of leave next week.
「休暇を申請しました」 says that leave has been requested. Approval must be checked separately, and the word alone does not establish whether a particular leave arrangement is paid.
Separating calendar days and personal leave
「休日と休暇を合わせて、一週間休みます」 says that the speaker will take a week off by combining regular days off with leave. Both kinds of days can belong to one break.
The broader word 休み is also common in everyday speech. Distinguishing 休日 and 休暇 helps when you need to clarify whether you mean a day off or a leave request.
Does 「休暇中です」 mean that today is a public holiday for everyone?
Show the answer and why
No. It means that the person concerned is on leave. The sentence does not tell you whether other people or institutions have the day off.