Hydrometers are based on Archimedes Principal. “Archimedes' principle indicates that the upward buoyant force that is exerted on a body immersed in a fluid, whether fully or partially submerged, is equal to the weight of the fluid that the body displaces and it acts in the upward direction at the center of mass of the displaced fluid.”
What that means to us as brewers is when the sweet wort is measured with your hydrometer the thicker it is, the higher your hydrometer floats. The more malt extract you use or grain you mash, the higher the reading. After the yeast works its magic and converts the sugars into alcohol, the reading will be lower. These two readings are known as OG, original gravity and FG, final gravity.
So what can we use these reading for? First if you are using a known recipe it may have an expected OG and FG. You can see if your beer matches the expected reading. Second you can figure out the alcohol content after the beer is done fermenting. Alcohol by volume or ABV can be figured out by subtracting the FG from the OG then multiply that by 131.25. So a beer with an OG of 1.050 that finishes at 1.010 would be 1.050-1.010=.04x131.25=5.25%ABV. There are also a lot of online calculators to do the math for you. Lastly you can determine if your beer is done fermenting or stuck. Simply measure the gravity each day for about three days. If it does not change, it is either done or you have a stuck fermentation. A typical FG is around 1.010 however some really big (high alcohol) beers may not reach that low.
How do we use a hydrometer? First when you get your hydrometer you may want to calibrate it. Don’t use the plastic tube it came in. They are not wide enough so the hydrometer may stick to the side and give a false reading. Buy yourself a hydrometer flask. They are cheap and will stand up by themselves. Place the hydrometer in it and fill it with 60° preferably distilled water. It should read 1.0. If not you will know how far off your reading is and adjust accordingly. If it is way off, return or exchange it for a new one. When you read your wort it should also be at 60° but there will be a temp adjustment chart with it. EDIT: Some hydrometers may have a different calibration temp.Check the card that comes with your new one.
When you take a reading of your wort, the easiest way to get a sample is to use a wine thief. They are just dunked in and have purge type valve that allows the wort in but not out until you press the valve against the inside of your hydrometer flask. Next spin the hydrometer or pour the sample back and forth between two glasses to remove any bubbles that may float the hydrometer too high. Then read the gravity at the meniscus. Wort will cling to the hydrometer and climb up a little. The meniscus is where the flat line is, not the top where the wort clings up higher. Don’t return the wort to the fermenter. It is not worth the risk of infection. Drink it or just toss it. Sanitize all of the equipment first!