Stars move against the background of the sky just as the planets do. (How can they move against the background, since they themselves form the background? Some are near to us, others far, others very far. The very farthest are the 'background', and the nearest are the only ones whose motion is big enough for us to discern. Even for them it is very small.)
That part of the motion caused by the star's motion through space is called the 'proper motion,' and is a few seconds per year for the fastest.
That motion caused by the motion of the viewer (i.e. the earth) is the 'parallactic motion', and causes a periodic (the period is one earth year of course) repetitive back and forth change in the star's position, typically a few tenths of a second, or smaller, in size. One half of the angle a star seems to move each six months due to the earth's motion around the sun is called its 'parallax' or 'parallax angle'.
A new unit of distance, the parsec, is used for convenience. A star whose parallax is one second is said to be one parsec distant; one whose parallax is 1/2 second is two parsecs distant, etc. A parsec thus defined comes out to be about 3 times the distance light travels in a year. A PARSEC IS A DISTANCE!!
The proper motion causes the arrangement of the stars in the sky to change. The constellations are not permanent when measured over tens of thousands of years.
The parallactic motion is our key to the distances to the stars. All celestial distance measurements are built upon this one.
Radial (along the line of sight, or along the line joining us and the star)
motion of a star will not produce any proper motion. It is detectble through
the Doppler effect in the star's spectrum.
Proper motion can be used to find tangential velocity (motion perpendicular to the line of sight) if a star's distance is known. Tangential velocity combined with radial velocity gives 'space velocity' for a star.