Up to now we have assumed that all objects we look at are stationary, and that while the Earth moves around the Sun, the stars we observe stay where they are. That is far from correct; stars themselves move like everything else in the Universe. All these motions are typically in the order of tens to hundreds of km per second. Because of the vast distances to us, these motions are hardly, if at all, noticeable within a human life span.
We need accurate and systematic measurements to find the actual motions of celestial objects. Any star can move in any direction with respect to us on Earth. When measuring that relative motion we must distinguish between two components
The transverse velocity of stars is the component of their motion perpendicular to our line of sight. We call that proper motion. The other component is the radial velocity or the velocity with which the star moves away from or towards us. That latter component does not affect the position of the star in our sky.
The annual parallax we can observe from Earth cannot be separated from the proper motion of a star
We will always observe the combination of the two. The difference however is that annual parallax is periodic, it repeats with a cycle of one year, whereas proper motion is generally secular, it shows the movement of the star in one general direction.
Barnard’s star (also dubbed “Greyhound of the Skies”) is the star with the largest proper motion. It is one of the closest stars at almost 6 ly and it has a proper motion of just over 10 arcseconds per year. Its annual parallax is about 0.55 arcseconds.
The diagram shows observations of Barnard's star over a period of more than one and a half year. The periodic motion due to annual parallax is clearly visible. The fitted straight line shows the proper motion after the parallax has been removed from the data.
If we want to calculate the distance to Barnard’s star from annual parallax, we must first remove this proper motion from the observations. Its annual parallax of 0.55 arcseconds gives a distance of
1 / 0.55 = 1.8 pc or 5.9 ly.
The star's proper motion is 10.55 arcseconds per year.
A detailed account of measuring Barnard’s Star is given in this publication by R.J. Vanderbei.
The movie we linked to above shows in the latter part the proper motion of stars. Both the parallax and the proper motions shown are based on data from the Gaia spacecraft.
See the ESA page and download option of the movie here.