For the time around the winter solstice, the sun does never rises at the North Pole. That's right, it's dark 24 hours a day (although I've heard it's not pitch-black -- you do get a little sunlight, but it's like dusk.)
The Earth is tilted at about a 23-degree angle to the sun. When were around Dec. 21, the North Pole is tilted farthest away from the sun and the angle is great enough that the area stays in the Earth's shadow at all points of the day. Of course, in summer, it's the opposite, and the sun doesnt set.