Categories like "Noun" and "Adjective" can be said to be lexical categories. But what are categories like "subject" and "object" called?
Subject and object are names of grammatical function categories.
The grammatical function of a constituent in a sentence is determined by its relation to other constituents in the sentence. The subject, for example, is what the meaning of the predicate is predicated of, and it also agrees with the head of the predicate - the main verb, if there is one - in person, number and gender. Note that verb is not a grammatical function category but a syntactic category. The verb in a sentence realizes the grammatical function of head of the predicate (sometimes called the predicator), as can be seen in the syntactic structure tree below, where the grammatical function categories appears in blue. You can read a little more about this distinction here.