The alcohol content of beer needs to be measured for two prime reasons;   The amount of tax paid on a beer is proportional to the alcohol content and so people can see how the beer they're drinking affects their sobriety.

A hydrometer (often called a saccharometer in breweries) is a tall, thin glass bulb with graduations up the side of it.   Just as a person floats further out of the water in the Dead Sea than in a swimming pool, the hydrometer floats at differing heights out of the liquid it is in depending on the liquids density.   The hydrometer is calibrated against known liquids and the density of pure water is defined as 1.000 g m^-3 at STP (standard temperature and pressure).   Pure ethanol has a density of 0.789 g cm^-3.   A saturated sucrose solution has a density of about 1.800 g cm^-3.

Original Gravity and Specific Gravity

The gravity of a liquid is its density compared to that of pure water.   This removes the units from the readings.   Original Gravity (OG) is the gravity of the wort before pitching the yeast.   Specific Gravity (SG) is the gravity at any given moment in time during the brewing process.   Terminal Gravity (TG) is the gravity of the wort once the yeast has metabolised all the sugars and the fermentation has finsihed.

The OG will be higher than 1.000 depending on the beer recipe (around 1.075 for barley wines or 1.030 for light lagers) as the amount of sugars raises the gravity.   As fermentation proceeds the SG falls as the sugar content drops and as the alcohol content rises (remember ethanol has a gravity of 0.789).   The SG will approach 1.000 as the fermentation completes and a strong, light bodied beer will have a TG of just less than 1.000.

Alcohol calculation

(OG - SG) / 7.4 = alcohol %

This is an approximate calculation often used by homebrewers.   The constant of 7.4 is dependant the large drop in gravity and so will be more accurate for a barley wine than a light lager, but is still good enough for homebrew readings.