Is whose an adverb or adjective?

The word 'whose' is both an adjective and a pronoun.
The adjective 'whose' is an interrogative adjective, a word that introduces a question.
The pronoun 'whose' is an interrogative pronoun and a relative pronoun.
The interrogative pronoun also introduces a question.

The distinction between the interrogative adjective and the interrogative pronoun is that the interrogative adjective is placed before a noun to describe that noun. The interrogative pronoun takes the place of a noun.

Whose car is in the driveway? (adjective, describes the noun 'car')
Whose is the car in the driveway? (pronoun, takes the place of the noun that answers the question)

The relative pronoun 'whose' introduces a relative clause, a group of words that gives information about its antecedent.
Example: The person whose car is in the driveway is my brother.