The Monty Hall Problem is one of those things that demonstrates just how powerful a pull common sense has on the human reasoning process. The problem itself is easily stated: there are three doors and behind one of them there is a prize and behind the other two, nothing. You choose a door in hopes of finding the prize and then one of the other two doors is opened to reveal nothing. You are offered the opportunity to switch your guess to the other door. Do you take it?
Common sense tells you that switching wouldn't make any difference. There are two remaining doors, the prize is randomly behind one of them, why would switching yield any benefit? But as the video explains and this simulation shows, counterintuition prevails: you should switch every time.
America was introduced to the difficulty of the problem by Marilyn vos Savant in her column for Parade magazine in 1990.^{1} In a followup explanation of the question, vos Savant offered a quite simple "proof" of the always switch method (from Wikipedia). Let's assume you pick door #1, here are the possible outcomes:
Door 1  Door 2  Door 3  Result if you stay  Result if you switch 

Car  Goat  Goat  Wins car  Wins goat 
Goat  Car  Goat  Wins goat  Wins car 
Goat  Goat  Car  Wins goat  Wins car 
As you can see, staying yields success 33% of the time while if you switch, you win 2 times out of three (67%), a result verified by a properly written simulator. In his Straight Dope column, Cecil Adams explained it like so (after he had gotten it wrong in the first place):
A friend of mine did suggest another way of thinking about the problem that may help clarify things. Suppose we have the three doors again, one concealing the prize. You pick door #1. Now you're offered this choice: open door #1, or open door #2 and door #3. In the latter case you keep the prize if it's behind either door. You'd rather have a twointhree shot at the prize than oneinthree, wouldn't you? If you think about it, the original problem offers you basically the same choice. Monty is saying in effect: you can keep your one door or you can have the other two doors, one of which (a nonprize door) I'll open for you.
See also the case of the plane and the conveyor belt (sorry not sorry, I couldn't resist).

I was a religious reader of Parade and remember this column and the resulting furor very clearly. Re: the furor, it's so interesting to note in hindsight (being 16 and clueless at the time) how much of the response was men who clearly were trying to put the smackdown on a prominent, intelligent woman and they just got totally owned.↩