As stated in Stellar Motions, the motion of a star relative to the Sun is referred to as its space velocity, and is divided into its radial velocity toward or away from the Sun, and its tangential velocity perpendicular to the radial velocity (hence, in the plane of the "sky"), as shown in the diagram below:

Measuring Radial Velocities
     Radial velocity is measured in terms of the change in the distance from the sun to the star. If this is increasing (the star is moving away from us), the radial velocity is positive; if it is decreasing (the star is moving toward us), the radial velocity is negative. We cannot use the radial velocity to decide whether the star is "really" moving toward or away from the Sun or vice-versa; what it measures is the relative motion of the Sun and star. To measure some kind of absolute motion in space we would have to define a reference frame based (for example) on the average motion of stars in our vicinity. This would involve a tremendous amount of work, and as we learn more might prove to be no more useful.
     The radial velocity of a star is measured by the Doppler Effect its motion produces in its spectrum, and unlike the tangential velocity or proper motion, which may take decades or millennia to measure, is more or less instantly determined by measuring the wavelengths of absorption lines in its spectrum. This can be accomplished regardless of the star's distance from the Sun, providing that it is bright enough to observe its spectrum in the first place. The only way that the star's distance affects the measurement is that the further away it is the fainter it appears, and the longer it takes to collect enough light to observe its spectrum. For the small radial velocities observed for stars in our own Galaxy the percentage change in the wavelengths emitted by the star, compared to their normal wavelengths, is the same as the star's radial velocity as a percentage of the speed of light. For instance, if the wavelengths are 1/10th of 1 percent less than normal, the star is moving toward us (or we are moving toward it) at 1/10th of 1 percent of the speed of light, or about 100 km/sec (which is actually a fairly high value); if the same percentage change were observed, but at a wavelength longer than normal, the star would be moving away from us at that speed.
     Unlike proper motion, which can only be measured for nearby objects, radial velocities can be measured for any object, even the most distant galaxies ever observed. Because of the expansion of the Universe, distant galaxies may be receding from us at thousands or even tens of thousands of km/sec. In such cases the percentage change in the normal wavelength is not the same as the actual recessional velocity (a term used in place of radial velocity when the distance is so large that objects at that distance are always "receding" from us), and special relativistic calculations have to be used. As an example, in the NGC/IC/PGC catalogs on this site, if the recessional velocity exceeds 7000 km/sec (a little over 2% of the speed of light) I list a distance calculated from the percentage change in wavelength, then the more accurate distance calculated using relativistic corrections.

Topics to be added or linked to at a later date

Background Physics: The Doppler Effect
(Discussion of the Doppler Effect, and how it can be used to measure radial velocities; merely hinted at above.)

Caveats
(Discussion of correcting for the Earth's orbital motion)
(Discussion of correcting for perturbations of our motion, e.g. due to the Moon)
(Discussion of correcting for the elliptical shape of our orbit, with a notable historical oversight)

Radial Velocities in Multiple Star Systems
(Discussion of variable radial velocities, and their causes: lead-in to Spectroscopic Binaries)
(discussion of using radial velocities to determine the masses of stars, clusters of stars, and galaxies)

Radial Velocities in the Universe
(Discussion of the radial velocities of galaxies: lead-in to The Expansion of the Universe)