Specific gravity is a fairly important concept in the world of beer, especially to brewers and home brewers (and the pedantic drinker). The specific gravity of a beer, both before and after it's fermented, reflects it's flavor, body, color, and alcohol content.
As other people have mentioned, specific gravity refers to density of something, compared to the density of water. The main ingredient of beer is water itself. The second most important, the sugary malt, is denser than water, do the unfinished beer – wort – is denser than water (there aren't enough of the other ingredients, yeast and hops, to affect the specific gravity much). By knowing the specific of the wort (called the original gravity), you can determine how malty it is, which will also tell you something about it's flavor, body, and color. In general, the higher the original gravity, the darker, the heavier, and the stronger the beer will be. For example, light beer will usually have an original gravity of between 1.020 and 1.040. Most beers fall into the range 1.040 and 1.050. Stronger, darker beers live at 1.050 and 1.075. Anything about 1.075 will be a might beer indeed.
The original gravity of a beer will also tell you how much alcohol the finished brew can have. If the wort has a high gravity, there is more sugar for the yeast to process into delicious, nutritious alcohol. That's why the original gravity of a beer is sometimes called potential alcohol. Of course, fermentation can be (and often is) stopped before all the sugar is fermented. Some sugars are also unfermentable, and will hang around no matter how long you give the yeast.
As the wort ferments, yeast will consume the sugars and turn them into alcohol, releasing carbon dioxide in the process. The carbon dioxide float out of solution, making the beer less dense, and therefore lowering the specific gravity of the brew.
Then specific gravity of the finished beer is also important. The ending gravity will tell you have much malt or sugar remains in the beer after it's been fermented. The higher the gravity, the more body the beer will generally have. The difference between original gravity and ending gravity will also tell you how much sugar was converted to alcohol, and therefore what the final alcohol content of the finished beer is.
And here you thought specific gravity was boring.
Another insteresting fact no one mentioned: specific gravity is unit less. Neat!