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Proper motion (PM) of a star is its change in position within the celestial sphere over time, corrected for the effects of Earth's orbit. Its proper motion at a given time can be given as a 2D vector located in the sky, indicated by two coordinate deltas (e.g., in arcseconds) per year, and the proper motion's speed can be quantified as angle per year, which is the magnitude of the vector. These are generally somewhat consistent over time, with a no more than a very slow change over the passing years. If you measure the precise position of a star at two instances precisely one year apart, the angle in arcseconds between the two measured positions is (the magnitude of) its proper motion. Precise position measurements within the celestial sphere are relative, and the determination of proper motions uses measures of the changes in angles between the star of interest and more-distant stars that show no proper motion relative to each other. A star's proper motion is an indication of its motion in relation to the Sun, but does not fully characterize that motion, which also requires knowing the distance to the star and the star's radial velocity.
Common proper motion is such motion when it is shared by two stars, evidence they are a binary star or co-members of a stellar association.
The term HPMS stands for high proper motion star. I don't know any standard criteria, but a proper motion of 0.2 arcseconds per year would be of interest, and even 0.1 arcseconds per year. The catalogs Luyten Half-second Catalog and Luyten Two-Tenths Arcsecond Catalog are classic efforts to catalog such stars. Such high proper motion is a sign that the star is relatively near.
When proper motion is quantified as a vector (citing two angular distance components), typical is to cite the change in declination along with a second angular distance consisting of the change in right ascension expressed as an angular distance along the circle of latitude (which would be along a curved line). The latter consists of the right ascension expressed as its angular distance around the equator times the cosine of the declination, which given typical annual proper motions, is virtually the same as the angular distance along a great circle, and the motion's angular-distance-per-year (vector magnitude) produced by these two using the Pythagorean theorem is hardly less accurate. Note that I've seen proper-motion citations (particularly in Wikipedia) that term these two quantities declination and right ascension, not specifying exactly what is meant by right ascension: when such a proper motion component is expressed as an angular distance (e.g., in degrees or arcseconds), it likely incorporates the cosine of the declination in this manner.