Hydrometers are the vessel through which your beer speaks to you. On brew day, you take the original gravity reading to know how much potential alcohol your beer will have, and also to calculate your mash efficiency if you are doing all-grain or partial mash. Later on, you take a reading to find out how far it's gone - this is a test reading - and then 3 days later you take another test reading to see if it is still changing. If so, you wait another few days or week or whatever and repeat. If not, congratulations! Your beer has just spoken to you to let you know it is at its final gravity and ready for the next phase of its life - secondary fermentation, bottling, kegging, or whatever your recipe calls for. The difference between your original gravity and final gravity can be used to calculate your alcohol content.
In a very literal sense, it tells you how the density of the liquid in your fermenter compares to the density of pure water at the temperature that it was calibrated for. For mine, 60*F water reads 1.000.
It will tell you how much residual sugars are left in the beer, and generally the closer your gravity gets to 1.000, the thinner the body. Higher FG #'s will usually have a thicker body - might taste "chewy" or, if you add a lot of unfermentables (high amounts of crystal malts, lactose, etc) it may even be cloying. For example, a stout with added maltodextrine and a little lactose will have a sweetness from the lactose and a chewiness from the maltodextrine - it will also have a higher final gravity than a crisp american light lager by a long shot.