Alcoholic proof is a measure of how much ethanol, or "grain alcohol," is in an alcoholic beverage. The proof number is twice the percentage of the alcohol content measured by volume. Therefore if a liquor is labeled "80 proof," it is 40% alcohol, by volume, and most of the other 60% is water. If you mix 180 proof whiskey half-and-half with water, your drink is 90 proof.
This system dates back to when spirits were graded with gunpowder, a solution of water and alcohol "proved" itself when it would ignite a pinch of gunpowder and was defined as 100 degrees "proof" (100°). This has since been found to actually be at 57.15% ethanol, but the American system was later changed when gunpowder was no longer used for testing and laboratory tests replaced it. (Note: The British proof system is still based on the original definition.)
Alcohol is produced by yeast during the process of fermentation (and the other product of fermentation is carbon dioxide, which is the gas that can make beer bottles explode or blow their tops off). The amount of alcohol in the finished liquid depends on how much sugar there was at the beginning for the yeast to convert into alcohol. In beer, the alcohol is generally 3% to 12% (6 to 24 proof) and usually about 4% to 6% (8 to 12 proof). Depending on the strain of yeast, wines top out at about 14% to 16% (28 to 32 proof), because that's the point in the fermentation process where the alcohol concentration kills the yeast. Interestingly, very few microorganisms can live in alcoholic solutions. The main three are yeast, Brettanomyces, and Acetobacter. In what is essentially disinfection, yeast keeps multiplying as long as there is sugar to "eat", gradually increasing the alcoholic content of the solution and killing off all other microorganisms, and eventually themselves. There are "fortified" wines with a higher alcohol concentration than that because stronger alcohol has been mixed with them.
Stronger liquors are distilled after fermentation is complete, to separate the alcoholic liquid from the remains of the grain, fruit, or whatever it was made from. The idea of distillation is that when you heat a mixture of liquids, the one with the lowest boiling point will evaporate (or "boil off") first, and then the one with the next lowest boiling point, and so on. The catch is that water and alcohol form a mixture (called an azeotrope) that has a lower boiling point than either one of them, so what distills off first is that mixture that is 95% alcohol and 5% water. Thus a distilled liquor can't be stronger than 95% (190 proof); there are other techniques for separating liquids that can produce 100% ethanol (or "absolute alcohol"), but they are used only for scientific or industrial purposes. 100% ethanol doesn't stay 100% for very long, because it is hygroscopic -- it absorbs water out of the atmosphere.