So the OG/FG readings from a hydrometer tell you two things, 1) indicates when the beer is done fermenting, and 2) what the alcohol content of the brew is.
Here's how to do, it's extremely simple:
1) When you've finished making the wort (and preferably before you pitch the yeast - although after isn't tragic), take a sample of the wort in a cyclinder - often the tube that the hydrometer comes in will work too. Set the hydrometer into it. Even though it's buoyant, be careful not to drop it in hard - hydrometers are quite fragile. You can leave it sit for a while as quite often there will be foam blocking you from taking a reading. Once the foam clears make a note of where the wort line is on the hydrometer. For example, you might get a reading of 1.052. Make a note of this - it's your original gravity (OG). Do not return the sample to the fermenter - drink it instead or pour it out.
2). When you suspect that your beer is done fermenting and ready to bottle, take another sample into the cylinder and perform another reading exactly as above. This is your final gravity (FG). Again, drink the sample. Wait a few days. Take another sample and do it again. If this reading is the same as the previous FG reading then you can be confident that your beer is done fermenting and is ready to bottle/keg. If the FG reading has dropped a couple of points then this shows that your beer is still fermenting and bottling would be dangerous at this point. Wait a number of days and go through this process again until you get a stable reading. Experienced Brewers will sometimes forego taking multiple FG readings, as we typically are aware of what to expect for a FG reading and seeing it once (or sometimes not at all) we go ahead and bottle when we think it's right. That said, we are also known to be wrong on occasion and be susceptible to exploding bottles. It's highly highly highly recommended that if not always, you do get a lot more experience in brewing before you limit or forego FG readings before bottling. Exploding bottles are no joke and there's more than one story on this forum of people dying from broken glass.
3). To know how much alcohol is in your beer, plug your OG and FG numbers into an online calculator, such as the one below. For example, if your OG is 1.052 and your FG is 1.010 then your beer is 5.5%.
http://www.brewersfriend.com/abv-calculator/
What's the science behind this?
Briefly - Wort is essentially water and sugar. The sugar comes from the grain. The water, you added yourself. That sugar-water is going to have a certain density, depending on how much sugar is in it. It'll be heavier than np lain water. The OG reading will tell you just how dense with sugar that water is. Then you add yeast and the yeast eat most of that sugar and convert it into alcohol and carbon dioxide. In doing so, they remove the sugars from the wort and change its density, bringing it closer back down to the density of plain water. The yeast will typically east most of the sugars, but not all. Some are 'unfermentable'. When you suspect the beer is done and you take a FG reading, you'll see how much sugar is left in the beer. Assuming you've arrived at a true FG (ie. no more sugar will be eaten by the yeast), then the difference between your OG reading and your FGreading will tell you how much sugar has been converted into alcohol, and hence the alcohol percentage of the beer. Getting a stable FG reading over several days will also tell you that the yeast have finished their work and it's safe to bottle. It would be unsafe to bottle beer that isn't done fermenting because of the CO2 produced during fermentation. If you seal a bottle and yet more co2 is being produced then the pressure inside it will mount and if enough, it will eventually explode. Glass shrapnel can be very dangerous.
