Red shift what happens when an object radiating light moves away from you really, really fast, in the same way a car's horn/clunking sounds lower when it goes away. Gorgonzola did a great writeup on Blue Shift which is the exact opposite, when the object is approaching.
Interestingly, the use of this shift by Edwin Hubble (1889-1953) was what led to the proof of the expanding universe as well as many very basic parts of Cosmology today: the age, condition, and fate of the Universe. An American astronomer had earlier noticed that the Andromeda Galaxy was receding away. Einstein had initially asserted an immobile Universe, but was corrected by Willem De Sitter.

All stars (in all directions) were moving away. Now, you can imagine one of a few things is going on. Either we are at the center of the Universe - unlikely - or the Universe is expanding. Here's the logic.

Imagine you have a big, red balloon, deflated. Draw a bunch of dots all over it. You can imagine that the distance between the dots will be on average, say, a centimeter or so. Start blowing it up. The dots will all start to spread apart. From any one dot's perspective, the other dots will appear to recede. That's what's basically going on with the Universe, and Hubble showed it because of the red shift. Also, the farthest dots will appear to recede faster, that is, really far stars are going to have really high red-shift.

Hubble, while all these crazy notions of an expanding Universe wer going on, during the 1920's and 30's, proved that there was a correlation between how far some stars were and how much they were red shifted. He convinced Einstein once and for all on the expanding Universe.

But that's not really what makes him or red-shift cool

The correlation between distance and red-shift was strong enough that Hubble said that the distance of an object could be calculated from its red-shift. The equasion goes like this:

Red Shift x Speed of Light
The Hubble Constant = Distance

The bad news is that we don't ... quite ... know what the Hubble Constant is, yet. It's somewhere between 50 and 100 kilometers per second for every megaparsec in distance, km/sec/Mpc. This means that a galaxy 1 megaparsec away will be receding from us between 50 and 100 km/sec, while another galaxy 100 megaparsecs away will be receding at 100 times this speed (bloody quick). So essentially, the Hubble constant sets the rate at which the Universe is expanding. Cool.

This brings us to Hubble's Law. v = H*r. V is how fast the object's going away. H is Hubble's (sort of) Constant. r is the velocity away from here.

Another cool thing about Hubble's Li'l Law: the inverse of the Hubble constant has units of time. By substituting in kilometers for Mpc in the Hubble constant, we find that upon inverting H we get a quantity with units of seconds (kilometers canceling out in the denominator and numerator). For a Hubble constant of 100 kilometers per second per Mpc, we get 3 x 10^7 seconds, or about 10 billion years. For H=50 kilometers per second per Mpc, the time scale is 20 billion years.

By figuring out this whole red-shift crap, we calculated the age of the Universe! Go Hubble! And you wondered why there was a big mirror (which was made at UA) in the sky named after him.

Sources: Pima Community College AST 104 Class
http://spaceboy.nasda.go.jp/note/kagaku/E/Kag13_e.html
http://astron.berkeley.edu/~mwhite/darkmatter/hubble.html