I was lucky enough that when I purchased my first beer brewing kit that it came with a basic hydrometer. I knew nothing about hydrometers but it seemed to me that it was a device that was used for measuring alcohol.
I guess I was partly right. A hydrometer measures the density or specific gravity of a liquid. The first beer kit that I used had asked you to obtain the original gravity and final gravity. My hydrometer showed a gravity setting, balling, and potential alcohol. To be honest with you I've never used the balling setting and I only occasionally look at the potential alcohol reading to get an idea if I'm course when I'm brewing my latest beer. Hmmmmm..... about 4 more weeks and my American Cream Ale from Brewers Best will be ready for summer drinking.
Ok.. you simply sanitize the hydrometer and the tube which holds the beer that you will be testing from and you are ready to go. Once I brewed my beer and was adding it to the plastic bucket (or carboy) I filled the tube up about 3/4's of the way and spun the hydrometer inside of the tube. I carefully took a measurement (let some foam dissipate) and had my original gravity. Then along the way after all fermentation was done and just before bottling, I took another hydrometer reading and wrote down my final gravity.
Here is the formula that I use to determine alcohol by volume (ABV).
Original Gravity - Final Gravity * 131.25 = Alcohol By Volume