A grammarian is someone who studies grammar and knows it in a serious way — how a language puts sentences together, what the rules are, and where the rules leak. In a university this can be a real job: describing the system, arguing about a construction, writing a grammar. In ordinary talk it can be slightly teasing: a grammarian is the person who cannot hear a sentence without noticing the join. Though actually, knowing grammar well and being unkind about other people’s English are not the same thing; the word itself only names the study, not the personality. It is a fairly formal, bookish word. You would not call a classmate a grammarian just because they remembered the past tense.