This is probably most important when dealing with relative clauses. Here's one casual definition of the use of "who" and "whom":

If the person described by a relative clause is the object of the verb in that relative clause, that person must be described with "whom".

If the person described by a relative clause is the subject of the verb in that relative clause, that person must be described with "who".

Examples:

"The guy who won the fight last night is a great boxer." (The guy WON the fight)

"The guy whom Mayweather beat last night is no longer undefeated." (Mayweather BEAT the guy)

"The teachers who work at Veritas are some of the best in the country." (The teachers WORK at Veritas)

"The professors whom I met at the conference don't approve of my punctuation." (I MET the professors)

A word of caution here, though: "whom" is a dying word, and many respected newspapers and periodicals have stopped bothering to use it "correctly" in these type of situations. Everyone seems to agree that 'whom' must be used as the object of a preposition ('of whom', 'to whom', 'for whom', etc.), but plenty of forward-thinking writers and editors have stopped bothering to perpetrate sentences like "The guy whom I know is coming to the party", which sounds incredibly stuffy and pompous.

At some point, usage trumps grammar and rules are forced to change, and "whom" in a relative clause is in that position now. (For example, if you want to be a real stickler for precedent, you should say something like "The guy yclepped Matt" or "Our teacher which that hight Matt" - after all, that was correct in the 14th century, and would still be if those medieval slackers had kept their English straight! )

The GMAT does have some funny notions of grammar, but the test seldom, if ever, forces you to produce phrases like "That girl whom I like", so I wouldn't sweat it. Just remember to use "whom" as the object of a preposition and you should be fine.