During the span of a human lifetime most stars appear absolutely fixed in position relative to each other. If you were to compare a photograph of the stars' positions on the day you were born to one taken on the day you die, virtually every stellar dot on the photographs would be in exactly the same position. But there are a few stars that would exhibit small changes in their position, and almost all of them would have noticeable changes if we increased the time between the images to thousands or tens of thousands of years.
The first person to note the motion of the stars was Edmund Halley. In 1718, while trying to more accurately determine the precession of the Equinoxes by comparing modern observations of the stars to the eighteen centuries' earlier observations recorded in Ptolemy's Almagest, he noticed that three stars — Arcturus, Sirius and Aldebaran — had differences in their celestial latitudes of 33, 22, and 20 arc-minutes respectively. Precession does not change the celestial latitudes of the stars, so the change was either an error in the older measurements or a change in the position of the stars, and Halley felt it more likely that the stars had moved than that the errors in their positions were so large.
As it happens, Halley's assumption wasn't entirely accurate. We now know that Arcturus has a motion of 2.3 arc-seconds per year relative to its neighbors, or more than a degree in the time interval considered by Halley, so its apparent change in position was certainly real. But for Aldebaran, modern estimates of its motion explain less than half of its variation in celestial latitude, so its difference in position was due more to measurement errors than any actual motion. Still, Halley's comments led to the realization that the stars do have motions relative to each other, and although most such motions are very small in any given year, over tens or hundreds of thousands of years they can produce remarkable changes in the appearance of the constellations, as shown in the illustration below. The top image shows the Big Dipper today. The center image shows how the stars in the Dipper are moving. Alkaid, at the handle end of the Dipper, and Dubhe, the upper "pointer" at top right of the bowl of the Dipper, are moving in one direction, while all the other stars are moving in the opposite direction. As a result, in 50,000 years (as shown in the bottom image) the shape of the Dipper will be noticeably different.
Top: The Big Dipper today
Middle: Motions of the stars in the Big Dipper.
Bottom: The Big Dipper 50,000 years from now.