"Ohayou" consists of [o-prefix + hayou]. "O-prefix" makes it politer. "Hayou" is the archaic form of "Hayai"="early" (an i-adjective). "Gozaimasu" is the polite form of "gozaru", and "gozaru" is the archaic form of "aru/dearu/da"=the verb "to exist/to be" or a copula "be". In modern Japanese, adding "gozaimasu" or saying "gozaimasu" in place of "desu/da" makes the whole phrase or sentence sound more polite.

the primary form of gozaimasu is gozaru ,and it is a kind of honorific to show your respect and politeness to others~so it doesn't have a exact meaning in this greeting~and when u greet your family and your friends who are very close to u~ you needn't to use honorific~ so the gozaimasu can be omitted~