One unit of alcohol is defined as equivalent to 8 grams of pure ethanol. Measurements of the alcohol content in drinks are commonly given as percentage alcohol by volume (AbV), also known as the Gay-Lussac system (in the USA, the proof system is commonly used for spirits, where the proof of a spirit is twice the AbV). The AbV expresses the volume of alcohol as a percentage of the total volume of the liquid. The density of ethanol at 20 degrees Celsius is 0.7893 g/ml. This means that one unit of alcohol is approximately 10 millilitres at room temperature, which makes calculations quite simple (at least in the metric system).

To calculate the number of units in a drink, use the formula

P * V / 1000
where P is the strength of the drink as a percentage alcohol by volume, and V is the volume in millilitres. Americans can use
P * Voz / 33.8
where Voz is the volume in fluid ounces.

Examples of this are:

Spirits, British measure (25ml) 40% AbV: 1 unit
Lager (1 UK pint/20 oz/568ml) 4.2% AbV: 2.4 units
Bottled beer (330ml/12 oz) 5.4% AbV: 1.8 units
Wine, large glass (175ml) 12% AbV: 2 units
Bottle of spirits (700ml/25oz) 40% AbV: 28 units

Approximately one unit of alcohol is metabolised by the body per hour. One gram of alcohol has 7 calories energy, so one unit will equal 56 calories. This does not take into account the sugar content of many alcoholic drinks. The recommended intakes according to the British government are 3-4 units per day for men, and 2-3 units for women (except during pregnancy); these figures vary slightly according to different estimates.

In most Western countries, including the USA and the United Kingdom, all alcoholic drinks must be labelled with the percentage alcohol by volume. However, in most places there is no legal requirement to display the number of units in an alcoholic drink, which is in practice the more useful figure.

Sources:
http://www.tastings.com/spirits/
http://www.nutrition.org.uk/Facts/energynut/alcohol.html
http://www.hebs.scot.nhs.uk/Learningcentre/obesity/4-2h.htm
http://www.atf.treas.gov/
http://www.ias.org.uk/pressmar02.htm

lj informs me that the proof system for measuring the alcoholic content of drinks is different in the UK from the US system; in Britain 100 proof is 57% ABV, which allegedly is the lowest concentration of alcohol that can be poured on gunpowder so that the powder can still explode. (Source: Plymouth Gin website at http://www.plymouthgin.com/index.cfm?articleid=98)