A Hydrometer is a weight with a calibrated displacement, used for measuring the Specific Gravity of a liquid. Brewing hydrometers are calibrated to the density of water (1.000), and typically have what are called "Dual" or "Triple" scale markings.
A Dual Scale will show Specific Gravity and Brix. A Triple Scale will show Specific Gravity, Brix, and Potential Alcohol Percentage (based on the assumption that your final gravity reaches 1.000).
Hydrometers are also available for Scientific or Distilling purposes, and will have scales calibrated to their specific uses. This article specifically discusses brewing hydrometers.
A hydrometer measures the density of the liquid it is submerged in against the density of pure water at a specific temperature - usually 60°F or 68°F - your hydrometer should be clearly marked, or at least come with documentation.
If the liquid is denser than water, the liquid will exert more force against the volume displaced by the bulb of the hydrometer, and less of the hydrometer will be allowed to submerge. If the liquid is less dense than water, the displaced volume will exert less force against the bulb, and the hydrometer stem will be further submerged. The markings on the stem are a ratio of the density of the liquid being tested to the density of pure water. If the reading is 1.112, the liquid is a multiple of 1.112 times denser than water. This reading is called the specific gravity.
Sugar is denser than water, so a solution of water and sugar has a higher density than pure water, and causes a suspended hydrometer to ride higher in the solution.
It is worth noting that alcohol has a lower density than water, and will introduce inaccuracy into readings. While this is true, it is not significant enough to matter.
In short, one uses the specific gravity to determine how much sugar is dissolved in the solution.
The potential alcohol of the beverage: if all of the sugar is fermented, how much alcohol will the beverage contain?
How much sugar remains in the final product.
Roughly how much sugar one may wish to add when back sweetening.
Gravity readings give you a snapshot of the sugar content of your fermentation. Single readings can be used to plan, and multiple readings can be used to gauge the health and status of your fermentation:
When planning a mead, you can determine how much honey will be required to reach your desired starting specific gravity (aka: original gravity).
When mixing the must, you can check that you've reached your desired original gravity.
If two readings are taken several days apart, you can determine whether your mead is still fermenting, stalled, or finished fermenting.
You can calculate how much sweetener is required for backsweetening.
You can measure when to arrest a fermentation, for the purposes of stabilizing at a predetermined level of alcohol and/or sweetness.
As always, start by sanitizing your equipment and workspace.
You will need your hydrometer, a hydrometer test jar, and a wine thief or turkey baster.
Place the Hydrometer into the test tube. Add a sample from your must or wort until it floats freely, without touching the bottom or sides of the jar. Once your Hydrometer is floating freely, place the test jar on a flat, level surface. Give it a little spin to dislodge bubbles and center the hydrometer, then take a close look at where the stem protrudes from the surface of the liquid.
Surface tension causes a small amount of liquid to climb above the surface, this is called the meniscus. Since the surface tension of liquids can change based on the make-up of the liquid, the height of the meniscus will also vary. Read the Hydrometer by looking through the meniscus, and draw an imaginary line level to the surface to the marking in the Hydrometer.
It may look simpler to take a reading by placing the hydrometer directly into the fermenter, but this is not best practice. Solids like fruit pulp or krausen can interfere with the reading. If the container is too shallow the hydrometer may not have room to float. As with discarding the sample, this also prevents unnecessary exposure to your must.
If you are confident in your sanitation, you can return your sample to your fermenter. This is extremely inadvisable when working with large batches, as it can introduce infections, contaminants, or unwanted oxygen. For the small batch home brewer, however, only a few readings can greatly diminish yield and increase head space. Use your best judgement.
Is my fermentation stuck? Take at least two readings, one week apart, if they are higher than your expected final gravity, and you get the same reading, yes, your fermentation is stuck. If you have different readings, your fermentation is still active, though it may be sluggish, a sign of under-nutrification.
How much alcohol is in my brew? There are several formulae and calculators available on the internet that allow you to approximate your alcohol content based on an original gravity and final gravity reading.
How much sugar do I need to make sure I hit X% ABV? Use an online calculator like the GotMead Calculator to determine your target specific gravity.
My brew is drier than I want, how do I know how much sugar to add to backsweeten? See back sweetening!
Is my brew stable enough to bottle? If you bottle a mead that has not finished fermenting, pressure can build up and cause the bottles to explode! Taking gravity readings at intervals will confirm that fermentation is not at high risk of restarting. If you are inexperienced, however, read about stabilization as well.