In "John is a student", "is" is a linking verb which actually connects the subject (John) to its predicate nominative (a student). A linking verb does not express an action, unlike other verbs such as "learn" or "do". Instead, it describes a quality or state of a subject.
In turn, a predicate nominative completes a linking verb and provides more information about the subject. In your sentence, the predicate nominative (i.e. a student) provides more information about who the subject (i.e. John) is.
As a side discussion, take a look at these two sentences below:
1. He fails badly. (grammatically correct)
"fails" is a verb describing an action; "He" describes the subject doing the action; "badly" is an adverb of manner showing how he failed.
However, we do not write:
2. He feels badly. (grammatically wrong)
This is because "feel" is another linking verb. "feel" does not describe an action although it is a verb; instead, it describes the state of mind of a person.
The sentence should be written as "He feels bad." (Subject--> He / Linking verb --> feels / "bad" --> predicate adjective (since it describes the subject))
I hope this helps.