When is it grammatically correct to use whom instead of who?

To determine whether who or whom should be used in a sentence, it may be helpful to recast the sentence in non-interrogative form using a personal pronoun (he, she, etc.) to determine whether he or she (subjects), or him or her (objects) would be used in place of who or whom. For example, in determining:
*Who did you call? versus Whom did you call?

this yields:
incorrect — *You called she.
correct — You called her.

Since the former sentence is a solecism, the latter sentence, using whom rather than who, is correct.

In determining:
Who is calling? versus *Whom is calling?

this yields:
correct — She is calling.
incorrect — *Her is calling.

indicating that who is appropriate for this question.

Whom is the form used when prepositions are involved. Again this is analogous to personal pronouns, for which the objective form is also used after a preposition. For example:
To whom have you been talking? (Compare: You have been talking to him.)
For whom have you taken these marvellous photographs? (You have taken these marvellous photographs for him.)
With whom are you going to the cinema? (You are going to the cinema with him.)
He sent gifts to his granddaughter, of whom he was fond.
He sent gifts to his granddaughter, whom he was fond of.

The indirect object is normally formed with to, so is generally just a particular example of the usage with prepositions (to whom). In some situations, whom can also be used by itself to represent the indirect object. However this is only really used in contexts where the direct object is omitted, e.g. Whom will you tell? Where the direct object is expressed, to is generally included, even where an analogous sentence using a personal pronoun might use the pronoun as the indirect object, e.g. You will tell him a story but To whom will you tell a story?

The same rules apply to the declined pronouns whomever and whomsoever.
(wiki)