A hydrometer allows you to measure the sugar content of the wort or must. A triple scale hydrometer (which I recommend) provides two ways to measure the sugar content; specific gravity and brix. The third scale provides a figure for the potential alcohol.
To use the the hydrometer, you should also have a hydrometer jar to hold your sample. The hydrometer jar can be a plastic (clear) tube mounted on a base, these are inexpensive and available at most beer and wine shops. A graduated cylinder is another good option. On my hydrometer jar, I have two marks which I use as fill lines. One is a minimum fill line for measuring the original gravity, the other is the minimum to measure the final gravity of a dry wine. This covers my expected range or readings. You can add your own fill lines as well. A good starting point would be to place the hydrometer in the jar and add water. Add enough water to float the hydrometer off of the bottom of the jar. You now have enough volume of liquid to make most measurements. Remove the hydrometer and mark the jar where the water level is. For the final gravity of dry wines and meads, you may need an additional quarter to half inch of fill to keep the hydrometer off of the bottom of the jar. The hydrometer should come with instructions, or the shop owner can show you how to take a proper measurement.
Knowing the specific gravity (SG) allows you to monitor the progress of your ferment. With beer, there is a "general rule" that says your final gravity (FG) will be about 25% of your original gravity (OG). There is not such a rule with wine.
If you keep a log book of your batches, your hydrometer readings will give you a point of reference to compare current batches with previous batches.
To calculate the alcohol content, subtract the potential alcohol (PA) of the finished product from the PA of OG. The result is the percent by volume.