We talk in astronomy about two different kinds of redshift, but they both relate to the motion of a distant object relative to Earth.

When an object that emits light (of any frequency that we can sense) we can measure the radiation's wavelength. When that object and the Earth are moving relative to one another, then the wavelength changes because of that relative motion.

If the distance between the object and the Earth is decreasing, then the incoming radiation is said to be blue-shifted, because the intrinsic wavelength (the wavelength at the source of the emission) seems to be decreased.

If the distance between the object and the Earth is increasing, then the incoming radiation is aid to be red-shifted, because the intrinsic wavelength (the wavelength at the source of the emission) seems to be increased.

If the object is inside the Milky Way it is said to be "local" and the red or blue shift is due solely to the relative motion of the star or other source and the Earth. This is stellar redshift.

But, if the object is much farther away (the light from a distant galaxy, for example) then the redshift or blueshift is due mostly to what is called cosmological redshift. The source of that redshift is largely what is called the Hubble Flow: the stretching of spacetime we suspect is due to the expansion of the Universe.

At local scales, the actual movement of the object is the primary component of redshift, whereas at larger scales the component due to the expansion of the Universe predominates.

You often find vigorous discussion and disagreement over the causes of cosmological redshift. Regardless, it is observed.

One way we have all experienced this compression and stretching of wavelength is in the sound of a train whistle, or a police siren.  As the train -- or police car -- comes toward you the sound has a higher pitch because the wavelengths are compressed by the motion (blue shift),  As it passes you and starts to move away the sound has a lower pitch because the sound waves are now stretching out longer (red shift). 

Here's an article;

http://en.wikipedia.org/wiki/Doppler_effect

 And while the pitch of a whistle as heard by ear is subjective, if you knew before hand what the pitch of the whistle was, and you measured it's apparent pitch, you could determine if it was moving forward or away from you, and at what speed.

If you bend a stars light with a prism you will find shadows mixed in with the colors. The shadows are absorption lines caused by certain elements absorbing certain frequencies of light. Determining a doppler shift involves finding known elements by their shadow and seeing how far to the red or the blue they are shifted.

For instance say that the spectral imprint for hydrogen was in a certain place in our own sun, if we looked at another sun and found that same imprint and it was in the blue region, we'd know that start was moving toward us. If it were in the red though we would know it was moving away from us. The farther those signatures are off the center and into red or blue, the faster away or towards us it is travelling.