You are probably asking yourself, "What the hell is a hydrometer?!" A hydrometer measures the weight or "gravity" of a fixed quantity of liquid and compares it to what that same volume of water would weigh. Therefore, the expression of gravity is measured as some offset of 1.000. A common starting gravity for a beer might be 1.50 and can reach measures of 1.090 (or 50 to 90 points as brewers like to say) or higher. Finishing gravities end closer to the gravity of water (1.000), such as 1.020. It is however the difference between the two that is really meaningful for calculating alcohol.
Hydrometers measure the density of liquids by measurin buoyancy. Just as it is easy to swim in salt water than in fresh, so the hydrometer floats more in dense wort (unfermented beer) than it does in thin wort. The density of the wort is determined by the amount of fermentables present (sweetness) which determines how much fermentation will take place once the yeast is added.
By measuring how much fermentables (yeast consumables) are present before adding yeast, and in turn, comparing that with the amount of fermentables present after fermentation, the amount of fermentables consumed by the yeast can be determined. By knowing how much food these little yeasties eat, we can approximate the amount of alcohol present.