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, measured as an angle (e.g., arcseconds) per year. If you could measure the precise position of an object at two instances precisely one year apart, the angle in arcseconds between the two measured positions would be its proper motion. Actual precise position measurements within the celestial sphere are relative, and the practical means of determining proper motion uses measures of the changes in angles between the star and neighbor stars within the celestial sphere that are far more distant and show no proper motion relative to each other. A star's proper motion is an indication of relative motion between the star and 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.
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 find such stars. Such high proper motion is a sign that the star is relatively near.