This is used to mean senior or most important. The big school is Secondary School (as apposed to Primary school) or High School:
“Ah’m goin’ tae the big school in August.”
Big lassie is a child’s term of address to an older girl or young woman:
“Hey big lassie, gonnae gie’s ma baw back?” (baw->ball->football)
Big man is a friendly term of address used to someone the speaker regards as being taller than himself:
“Can Ah get a swatch at yer paper*, big man?”
*swatch at yer paper = can I have a wee look at your newspaper?