The Canonical Version:

You are shown three identical doors. Behind one of them is a car. The other two doors conceal goats. You are asked to choose, but not open, one of the doors. After doing so, Monty, who knows where the car is, will open one of the two remaining doors. He will always opens a door he knows to be hiding a goat. After he opens one of the doors and shows you the goat, he will give you the option of either sticking with your original choice or switching to the other door he did not open. You will then receive whatever is behind the door you open
This wording of the problem taken from Jason Rosenhouse, THE MONTY HALL PROBLEM, Oxford University Press (2009), pp. 35-36.

What should you do?
a) Stick with the first door you chose.
b) Switch doors.
c) It doesn't matter.

X

The Probabilities:

a) Stick with the first door you choose and have a 33% probability of winning the car.
b) Switch doors and you have a 66% probability of winning the car.
c) It does matter.

Why?

Before the game begins you know there is a probability of 33% that the car is behind the door you originally choose.
That means that there is a probability of 66% that the car is behind one of the two remaining doors you did NOT originally choose.
Monty will now show you which, of the two remaining doors, the car is NOT behind.
So if the car IS behind the two remaining doors, which will be true 66% of the time, he has shown you exactly where it is.

These statistics are borne out by the actual TV show, by classroom experiments and by computer simulations.
It seems that pigeons are better able to learn this lesson than we humans...

For further explanation, see Jason Rosenhouse, THE MONTY HALL PROBLEM, Oxford University Press (2009), pp. 35-36.