The apparent change of position of a star on the celestial sphere is called the proper motion of the star.

Angular Motion on the Celestial Sphere

Proper motion is usually denoted by the Greek symbol "mu", and is a velocity that is usually quoted in units of seconds of arc per year. The following image illustrates proper motion.

The proper motion of a star on the celestial sphere.

Barnard's Star

Proper motion is not large. The star with the largest proper motion is called Barnard's Star. It moves 10.3 seconds of arc per year. Since the moon subtends about 1/2 of a degree (which is 1/2 x 60 x 60 = 1800 seconds of arc) on the celestial sphere, it takes Barnard's star about 1800/10.3 ~ 180 years to change its position by the angular diameter of the moon. All other stars have smaller proper motions.

Proper Motion and Parallax

Were it not for the motion of the Earth around the Sun, proper motion would lead to a simple drift of the position of a star on the celestial sphere in a particular direction. However, because of the motion of the Earth on its orbit, there is a parallax effect for stars that are near enough to exhibit significant proper motion that causes the star to execute motion in a small ellipse on the celestial sphere over a period of a year (this ellipse is just a mirror of the actual motion of the Earth on its elliptical orbit).

The superposition of these two motions (a straight linear drift from the proper motion and elliptical motion from the parallax effect) then leads to a wavy path of the star on the celestial sphere. The following figure illustrates for Barnard's Star (Source).

The proper motion of Barnard's Star

The wavy motion is the parallax effect for an amateur telescope observation from Earth (notice the periodicity of 1 year). The straight line is the average path as determined by the Hipparcos satellite.

Here is a Table that lists the stars having the largest proper motion, as viewed from Earth.

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