What does gravity have to do with booze?

So, what does gravity have to do with booze anyway?  More specifically, what are original gravity and final gravity, and what do they mean?

Basically, these are measurements which relate to the sweetness, the dryness, and the alcohol content of a beer or wine.

Measuring actual sugar and alcohol content precisely and scientifically is a very complicated and expensive process.  By using gravity (more specifically, the original (OG) and final (FG) gravities) brewers and winemakers can estimate these measures.

Sugar is heavier than water, so by using a hydrometer, brewers and winemakers can estimate the sugar content of their juice or wort through the original gravity.  Water has a specific gravity of 1.000.  So, a hydrometer which rests slightly higher than it would in pure water (say, in juice or wort) would measure something like 1.050 or 1.060.  The higher the original gravity, the higher the sugar content of the juice or wort.

Alcohol, on the other hand, has a lower specific gravity than water.  As fermentation progresses, the specific gravity of the wort or juice drops as well.  Through their own and other brewers' and winemakers' past experiences, brewers and winemakers know what the expected final gravity for a given style of beer and wine is, based on the wort or juice, as well as the particular yeast strain.  During fermentation, they can use a hydrometer to check the "attenuation" or the degree of fermentation, of their product.  Each yeast strain, as well as grain or grape recipe, has a different expected or target attenuation.  That is, what percentage of the sugars will be turned into alcohol?

What does this mean for us as beer or wine drinkers?  Well, sweetness (as opposed to fruity flavors, which can come from esthers even in dry wine) is a measurement of the sugar remaining in a finished beer or wine.  Essentially, this is the result of two decisions the brewer or winemaker makes: how much sugar to start with (original gravity), and the degree to which the sugars will be converted into alcohol (attenuation), which affect the final gravity of a beer.

Final gravity is a direct correlation to sweetness, and is also a good measure of what the mouth feel of a beer will be like, more or less, since the heavier a beer, the heavier the body will be.

All other things being equal (original ingredients, or at least the original gravity of them), a higher alcohol beer or wine will be drier than a low alcohol beer or wine, as more sugar was converted to alcohol.  Meaning that if you want to make a sweeter beer or wine, there are three ways to do it.
1) You can add more sugar (malt/grain or grape juice) at the beginning of fermentation, or pull water out of your juice to concentrate sugars.  This will result in a higher OG, meaning more fermentable sugars.  Even after attaining the same degree of attenuation, this beer will have a higher final gravity, meaning both more sugar and more alcohol, than a beer with a more watered down, less sweet base.
2) You can stop fermentation, by either using a yeast strain with a lower alcohol threshold or lower expected attenuation, or by actually forcing fermentation to stop by removing yeast or changing the temperature or environment to stop fermentation.  This will create a lower alcohol, sweeter product.
3) You can add sugars during or after fermentation.  Depending on whether the goal is more alcohol and more sugar, or just a sweeter product, this can be done at various times during fermentation.  In winemaking, this is known as chapetalization, and is prohibited in certain regions and styles in old world winemaking.  If you add the same type of sugar (grape juice for wine, or malt/grain for beer), before fermentation is completed, you will wind up with a higher alcohol, sweeter product.  If you add the sugar after fermentation is completed, the ABV will remain steady, and the final gravity, or sweetness of the product, will increase.  Alternatively, you can add other types of sugar.  This is common in Belgian styles, where candy sugar, which is not fermentable by ale yeasts, is added to the wort to create a sweeter finished product, without limiting the fermentation of the sugars in the wort.