An adjective clause is a clause which performs the function of an adjective in a sentence. The adjective is a word which modifies a noun or pronoun in a sentence. For example, a tall man, (where tall is an adjective) etc.
The adjective clause may be introduced by a relative pronoun (whom, which, that, who etc) or relative adverb (where, when, after, before etc).
Ram, who is the monitor of our class, is intelligent.
She is the neighbor, whose dog is Ill-behaved.
Give me a reason why should I believe you.
Here, 'who is the monitor of the class' in the first example,
'Whose dog is ill-behaved' in the second example and 'why should I believe you' are illustrations of adjective clauses.
The reason is that in the first example, Ram is a noun and the clause explains the noun. In the second example, 'neighbor' is a common noun and the clause explains it. In the third example, 'reason' is again a common noun and the best part of the sentence explains it.
NOTE: Adjective clause is a kind of subordinate clause, which may also be called a relative clause.