이신가요 is a polite, honorific form of 이시다.
* 이시다 = honorific form of 이다 = are / is.
* -(으)ㄴ가요 = -(으)ㄴ가 (question ending) + 요 (politeness suffix).
* -(으)ㄴ가? = formal question ending which is softer and less direct than the normal -아/어요? or -ㅂ니까?
So you can think of examples such as:
- 휴가여서 좋아요? (polite, direct) => 휴가여서 좋은가요? (polite, less direct) => 휴가여서 좋으신가요? (more formal)
- 어디가 아프세요? => 어디가 아픈가요? => 어디가 아프신가요?
'시' is the "honorific" suffix that raises the subject of the verb (말한다 -> 말하신다, 몸이 아프다 -> 몸이 아프시다).
-(으)신가요 is mostly used in a formal setting, such as with someone you met for the first time.
'그대 오늘 어떤 기분이신가요?' is a stilted phrase that might appear in a magazine article. It is not in a conversation style.
It means "What kind of mood are you in today?": 그대(you) 오늘 어떤 (what kind of) 기분(mood)이신가요(are you)?