What is an example of when to use who or whom?

The pronoun 'who' is used as a subject pronoun.
The pronoun 'whom' is used as an object pronoun.

The pronouns 'who' and 'whom' are interrogative pronouns and relative pronouns.
An interrogative pronoun introduces a question.
A relative pronoun introduces a relative clause.

Examples:
Who is your new math teacher? (interrogative pronoun, subject of the sentence)
Mr. Smith who taught science is also teaching math. (relative pronoun, subject of the relative clause)

To whom should I give my completed application? (interrogative pronoun, object of the preposition 'to')
The manager is the one to whom you give the application. (relative pronoun, object of the preposition 'to')