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Short answer: When in doubt, use who. It's disconcerting to hear whom where who is expected, but the usage of who in situations where previously whom was standard has been increasing, especially in spoken usage.

Longer answer: The traditional rule is that whom was to be used in the "objective case". What this means in practice (it's even controversial whether English has cases), is that you try to answer the question: if the answer is he, she, they, I, we, etc., you use who. If the answer is him, her, them, me, us, etc., you use whom.

Examples:

• "The man who spoke yesterday…", not "the man whom spoke…" ("He spoke" is correct; "Him spoke" is not.)




• "Whom did you see?", not "Who did you see?" ("I saw him", not "I saw he".) The latter is frequently common these days, though.

The Language Log posts (1, 2, 3) linked in another answer, as well as William Safire quoted on the Wikipedia page, recommend avoiding whom or recasting your sentence if it seems necessary.

Someone using whom in place of who is likely to be interpreted as a hypercorrection from linguistic insecurity (and Geoff Pullum at the Language Log agrees), while using who in place of whom is, at worst, being too colloquial (and at best, being hip and cool!). Summary: it's good to know which is which and use them correctly, but when in doubt, use who.

"Whom remains in significant use following a preposition" but use in objective case is moribund. The Wikipedia article on "who" has a detailed explanation.

The death of "whom" has been tracked on Language Log over the years. For example, here and here.

More examples:

• "It's a made-up word used to trick students."


• "As far as I'm concerned, 'whom' is a word that was invented to make everyone sound like a butler."

Who and whom also happen to be relative pronouns.
Relative pronouns link noun phrases (NP) to relative clauses (RC).

Who is the subject pronoun, and it has its object form whom and possessive form whose. Who and whom refer to people only.

For the last half century or so who has been used more and more for both positions: subject and object. Whom, on the other hand, is used as an object or as the complement of a preposition in formal contexts.

• That is the repairmen who fixed your car.


• Your friend Alex and his wife Samantha, whom he courted for so long, are getting a divorce.

Which is another relative pronoun, but not used for persons. Even though I have seen several very good writers use and get away with it.

Whose is the easiest relative pronoun to use: it can be used for people, animals, and things.

This is an attempt to also formulate an answer to my own recent question which was marked as a duplicate:

How do I choose between ‘who’ or ‘whom’ when the subject pronoun is murky?

I am still working on this subject but what I have managed to grasp is the following;

'who' is a subject pronoun and 'whom' is an object pronoun

as explained by Professor Malcolm Gibson's article Who is correct? Yes, though it may depend on whom you ask! which I referenced in two of my answers on ELU regarding who/whom: therefore an example of a practical approach that I would suggest at present would be

1. Look at the verb that requires who/whom:
   
   if the verb needs a subject pronoun, select ‘who’.
   If the verb does not need a subject pronoun, see whether it is a transitive or an intransitive verb.




2. If it is a transitive verb, it needs an object pronoun: so select ‘whom’




3. If it is an intransitive verb or a linking verb, it does not need an object pronoun: so select ‘who’.




4. In more complex sentences having multiple verbs, the choice of who/whom depends on whether the clause as a whole takes a subject pronoun or an object pronoun.

Appendix: Explanatory notes

The so-called ‘simple substitution rule’ given in earlier answers works especially well in simple sentences, as does my own method highlighted above. Problems arise when the sentence does not obviously require a subject pronoun, and the concerned verb is moreover not obviously transitive, making it difficult to decide whether it needs an object pronoun.

As @Peter Shor pointed out in the comments, problems also arise when the sentence has more than one verb, in which case the problem is how to decide which verb ‘who/whom’ would apply to: to determine which, some element of contextual reading and deconstruction might be required.

Example: “These are the men who they believe conspired to rob a bank.” – Peter Shor

Decoding this sentence, “what do they believe?” – “they believe (that) these men conspired to rob a bank.”

Whom is not to be used after ‘these are the men’ because the object of the transitive verb ‘believe’ is not ‘these men’ but the object clause, “these men conspired to rob a bank.”

Now it is notable that changing any verb other than 'believe' will not change the choice of who/whom: not even if a transitive verb like '(they) saw', which makes ‘these men’ its object, were to replace ‘conspired’; as in

they believe (that) they saw these men rob the bank:

these are the men who they believe they saw rob the bank.

Why not ‘whom’, since they saw these men? Aren't these the men whom they saw rob the bank? Not exactly, because they only 'believe' so – and the object of the verb ‘believe’ is again not ‘these men’ but the object clause 'they saw these men rob the bank' – in short, ‘these men’ never being the object of ‘believe’, ‘whom’ is not to be selected here.

Modifying the use of ‘believe’ a little will, however, change the choice of who/whom:

They believe these men to have conspired to rob the bank.

‘These men’ having finally become the object of ‘believe’, ‘whom’ is now required here.

These are the men (whom) they believe to have conspired to rob the bank.

Thus we might conclude, as @Andrew Leach suggested in a comment on my recent who/whom question, that

(4) in more complex sentences having multiple verbs, the choice of who/whom depends on whether the clause as a whole takes a subject pronoun or an object pronoun [paraphrase].

Expert members please evaluate this method and post comments suggesting corrections/improvements to the practical approach I have highlighted in this answer.

If you want to sound very studious and correct (but also quite stuffy), use "who" when it's functioning as a subject and "whom" when it's functioning as an object.

Examples:

1. With whom are you going to the store?


2. Who is the boss around here?

I have to admit, while I still argue in favor of the subjunctive mood and some other finer, old-school, points of style, I've given up on who/whom and use "who" for everything, except in formal writing. I still use it properly in a few constructions (e.g. "...with whomever you wish"), but I mostly have given up on it as a lost cause, and maybe not a worthwhile cause to begin with.

Unfortunately, there is no single rule for using who vs. whom "correctly"; or at least, if there is one, it's a pretty complicated rule that can't be boiled down to just one easy test. (And as other people have pointed out, there are problems with the idea that the use of who in situations where the "rule(s)" call for whom can be characterized as "colloquial" or "incorrect".)

The "substitute he and him for who and whom, then rearrange the word order to see which sounds right" test that moioci mentions in the accepted answer is fine as a rule of thumb that usually points to the right answer, but I wanted to make it clear that it isn't always correct.

I would strongly recommend reading the Wikipedia article that nohat♦ linked to in his answer.

Here are a few of the relevant exceptions to the "substitute he and him for who and whom" rule that are covered there and in some other places on the web.

It is not standard to use whom predicatively in the same way as him

People often say things like "It was him", where the "accusative" pronoun him is used predicatively in a clause with the "nominative" subject it. But it is not considered correct to use whom as a predicate in sentences like Who was it? Who am I? Who are they?

The only situation where it might be considered correct from a traditional prescriptivist standpoint to use whom predicatively is when there is an "accusative" subject, which occurs in some sentences with a non-finite form of the copular verb be; for example, "Who(m) did you want him to be?" And in reality, sentences of this type are very rare, and who is often used instead of whom when the pronoun comes at the start of an interrogative sentence, so I doubt a descriptive grammarian would even recognize this as a possible exception to the general pattern of using who as a predicative complement.

Prescriptive guides are against using whom as the subject of a clause embedded in the relative clause, but descriptivist sources say that whom is fairly common in this context

Another issue is the use of who vs. whom in sentences that have a relative clause that itself contains another embedded clause.

The example given by Wikipedia is

Beethoven, who you say was a great composer, wrote only one opera.

The main clause of this sentence is "Beethoven [...] wrote only one opera", with "Beethoven" as the subject. The relative clause is "who you say was a great composer", with "you" as the subject. The clause embedded in the relative clause is "who ... was a great composer", with "who" as the subject.

Because "who" is the subject of the clause embedded in the relative clause, prescriptivist grammarians say the "nominative case" form who, rather than whom, should be used. This conforms to the "he-him" rule: we would say "you say he was a great composer" and not "you say him was a great composer".

However, in practice, many speakers/writers use whom in contexts like this, apparently following a different rule that goes something like "use whom when the pronoun comes before the subject of a relative clause".

There are some other questions on this site about this topic:

• The use of nominative "whom"




• "Whom" or "who" for object of a subordinate clause inside a relative clause

User F.E. left some comments beneath "The use of nominative 'whom'" that seem to accurately represent the descriptivist position, so I have copied them below (with minor formatting changes):

You might be interested in the topic of nominative 'whom'. It is often covered by a decent usage dictionary, such as Merriam-Webster's (Concise) Dictionary of English Usage. One such example, in MWCDEU entry "who, whom", page 782: "Aikman will always have a chilly relationship with coach Barry Switzer, whom Aikman believes is too soft when it comes to player discipline" -- Peter King, Sports Illustrated, 1997.

This topic of "who versus whom", where the word is the subject of an embedded content clause is discussed in the 2002 reference grammar by Huddleston and Pullum et al., The Cambridge Grammar of the English Language (CGEL), pages 466-7, where the nominative 'whom' is used by "Dialect B" speakers. CGEL states "That this is another place, however, where it is invalid to talk in terms of hypercorrection. . . . It has to be accepted as an established variant of the standard language."

And so, it seems to me that reasonable arguments could probably be made to consider that example in your post to be acceptable in today's standard English.

Perhaps I ought to explicitly mention that "Dialect A" usage (where speakers would use "who" in your example) is mentioned by the 2002 CGEL on page 467 as: "Dialect A, which selects nominative, has more speakers and is the one recommended by the manuals, but there is no reason to say that it is inherently better or more grammatically correct than Dialect B, which selects accusative: the dialects just have different rules."

Have a look at this:

1 the man + he robbed the bank > the man who robbed the bank

2 the man + we saw him yesterday > the man whom we saw yesterday

This is correct elevated style. "whom" tends to be replaced by "who"
and can be omitted altogether.