A hydrometer measures specific gravity (SG), which is the density of the liquid. Density is expressed in grams/liter. Water has a density of 1000 grams/liter. Wood and oil have less (800-900 grams/liter), that's why they float on water. Sugary water has a higher density than 1000.
Adding 10 grams of sugar to one liter (=1000 grams) of water will result in a 1% sugar solution (1 Brix). This solution will now have a density of 1004 grams per liter. This scale is somewhat linear for reasonable values: adding 100 grams of sugar to one liter gives a 10 brix solution (10% sugar solution), with a density or specific gravity of 1040 grams per liter.
If I give you a sugar/water solution without telling you how much sugar is in there, you can derive this by measuring the SG: e.g. If you measure 1040 grams per liter, you know this is a 10% sugar solution by weight. If you know how much liquid you have, you can derive the total amount of sugar as well. A hydrometer does just this: it tells you what the specific gravity is on one of the scales, and the equivalent amount of sugar/liter on another scale. If it reads 1.04 on the SG scale (a density of 1040 grams/liter), it will read 100 grams sugar per liter (or 10 brix) on another scale. The hydrometer is actually a hollow tube, and the lower the density of the liquid, the deeper it will sink. Careful calibration allows you to derive a density.
Now, you are planning on fermenting this sugary water. Every sugar molecule gets broken down into one ethanol alcohol molecule, and one CO2 molecule. These two latter molecules weight roughly the same (ethanol slightly lighter than CO2), so this gives a ballpark figure of 1 gram sugar becoming 0.5 gram alcohol. (0.52 to be exact). So if you turn all the sugar in your solution into alcohol, you know exactly how much alcohol there will be as long as you know how much sugar there was to begin with. Your hydrometer tells you this as well: It knows the sugar density, so it also knows what the alcohol density will become when all this sugar is fermented. This explains the third scale: The potential alcohol scale. This way, the density of the liquid (SG), the amount of sugar/liter, and the possible future amount of alcohol, can all be derived by a single device.
There are several caveats here: First of all, the potential alcohol level is only reached if all sugar is fermented. If your ferment stalls earlier, there will be remaining unfermented sugar, and hence a lower final ABV. The difference between the original and final gravity can be used to derive the ABV in these cases. It will be roughly the difference between these two potential alcohol values. Second, all this assumes that the water is at 20 degrees C, the scientific "room temperature". Colder/warmer water is denser/less dense, and the SG readings will be off. Your device usually comes with instructions, containing an equation to correct for different temperatures. Furthermore, these numbers assume that your solution consists only of water and sugar (and not just any type of sugar: sucrose). There are other sugars that behave slightly differently, and other soluble chemicals in your must that will also alter the gravity.
The mechanics of reading a hydrometer is explained very nicely by bernardsmith above. I would only like to add that the scales can be confusing: These are small numbers with lots of significant digits, and there is not much room on the little piece of paper inside. That's why they list only the significant digits: there is a line that says 1.000. But the next numbers are 10, 20, ... These are the final significant digits: If I end up at 10, it means the SG is 1.010 (the last two digits of the 1.000 become 10). Further down the scale you find 1.100, again followed by 10, 20, 30, ... Same story here, only the final two digits get changed from 1.100. So finding a 30 below 1.100 means a SG of 1.130. Note that the fourth digit is not written explicitly, but can be derived from the scale lines. These are often in 0.002 increments, not 0.001, so watch out there.