Different thicknesses primarily affect the types of sugars that are produced during the mash. (If you need a quick primer on what to expect of worts of different sugar compositions, look here.)
Here's a quick practical rundown of what you can expect from a thick and thin mashes, respectively. Thick mashes tend to produce a higher proportion of dextrins, which lend a fullness and sweetness to the finished beer. Noonan writes, "A thick mash (less than three-tenths of a gallon of water per pound of malt) induces the greatest overall extraction. A much thinner mash increases the proportion of maltose, and thus wort attenuation." For reference, 3/10 of a gallon is 38.4 fluid ounces, or 1.2 quarts.
Bottom line: A thicker mash will typically result in a more dextrinous wort. A thinner mash, on the other hand, will typically result in a thinner, more highly attenuated wort. As far as numbers are concerned, less than 1 qt/lb. would probably be considered thick. Likewise, more than 1.5 qt./lb. would probably be considered on the thin side.
You'll just have to wait around for Part 2 to learn about the mechanisms responsible for this difference.
References: Noonan, Gregory New Brewing Lager Beer (Brewers Publications, 1996) pg. 140-1