오늘이 며칠인지 알려주세요. = Please tell me what day it is today. [며칠이다 + ㄴ 지 = 며칠인지)
며칠 is a question word that comes from [몇 + 일] and it's usually contracted to "며칠."
The difference between 며칠 and 무슨 요일 would be that, 며칠 asks for what day of the month (1st, 3rd, 13th, 25th etc) whereas 무슨 요일 asks for which day of the week (Wednesday, Friday, Sunday etc.)