What is Proper Motion?
To understand a star's proper motion, you first have to understand something astronomers call the celestial sphere. Imagine, hypothetically, that there is an enormous sphere that surrounds the Earth. This sphere rotates once per day, and every star in the night sky is a point of light in a fixed position on the inside of this sphere. This sphere has a few properties, like north and south poles that stay in one position, and an equator that goes around its center. A star's position on the celestial sphere is measured by its right ascension and its declination, which are similar to longitude and latitude on a globe. These are measured in degrees, arcminutes and arcseconds. The equator of the celestial sphere is broken up into 360 degrees, with each degree measuring 60 arcminutes and each arcminute measuring 60 arcseconds. This means that one complete revolution on the equator of the celestial sphere is broken up into 1,296,000 arcseconds.
The star with the highest measured proper motion, Barnard's star, travels a total of 10.3 arcseconds per year on the celestial sphere. For thousands of years astronomers were unable to detect the proper motion of stars because the changes in their positions were so insanely tiny. This is caused by the fact that stars are so far away from the Earth. So while a star might be traveling through space at hundreds of kilometers per second, it could be hundreds of trillions of kilometers away from the Earth. Ancient Greeks suspected the existence of proper motion, but it wasn't actually measured until 1718 by the astronomer Edmund Halley. He noticed that Sirius, Aldebaran, and Arcturus had moved half a degree when compared to the measurements made by Hipparchus 1850 years earlier.
Today, modern technology makes it much easier to measure the proper motion of stars in the Milky Way Galaxy and a few other galaxies in our Local Group. Unfortunately, it is not yet possible to take accurate enough measurements to measure the proper motion of more distant objects in the Universe. Proper motion is most easily seen for objects that are close to the Earth and are moving tangentially to the Earth, not directly toward or away from it. In fact, this is a method used to find the stars that are closest to us in the galaxy.