First of all, the North Star (that dot that earthlings currently see in the night sky when looking North) is not actually a single star. The North Star, also called Polaris, is a multiple star system which actually consists of five different stars. Three of these stars are relatively close to each other and are in orbit around each other. The two others are very distant from these first three, and just appear at the same point in the sky by random chance because they lie on the same line of sight from earth. If viewed from another galaxy, these two other stars would not line up and would not appear to be a part of the main Polaris system. The brightest star of the main system is a supergiant, and the other two stars are smaller and orbit around it.
Secondly, none of the stars in the sky really move over the course of a single day. They are all stuck in place. (The stars do have movement, but these movement are measured in millions of years and not days.) The stars seem to all sweep across the sky every night because the earth is rotating. The earth rotates on its axis once a day. As a result, all of the stars in the sky sweep through great arcs and take about a day to return back their original location. The closer a point on the earth is to its axis of rotation; which cuts through the North and South geographic poles; the less that point moves. A person standing exactly on the North pole does not move at all over the course of a day. It's like spinning a basketball on your finger. The points on the basketball that are far from the axis of rotation (the sides) move very quickly. But the point on the axis (where your finger touches) hardly moves at all. That is why you can balance the ball on one finger when it is spinning. And because the daily movement of the stars in the sky is caused by earth's rotation, the closer a star is to the axis of earth's rotation (an imaginary line that extends straight up from the North and South pole, out into space), the slower it moves in the sky. There is nothing special about Polaris beyond the fact that it ended up being the one right now sitting closest to the earth's axis of rotation in the North. Because it is so close to the axis, Polaris moves very little in the sky throughout the night.
Thirdly, Polaris is not exactly lined up with earth's axis of rotation. There is very low probability that any star would end up exactly lined up with earth's axis. Polaris lies at a viewing angle that is 0.736 degrees away from exact North. Because the North Star does not lie exactly on earth's rotation axis, it actually arcs through the sky every night. The arc is just so small that humans can't see it. Furthermore, earth's rotation axis is not completely fixed. Because earth is not perfectly spherical but bulges at the equator, the sun's gravitational effects on the earth are not completely symmetrical. This non-symmetry creates a very small and very slow twist on the earth that is turning earth's rotation axis away from Polaris. In about 3000 years, the star named Gamma Cephei will end up much closer to earth's rotation axis than Polaris, and will therefore be the new "North Star".