In Britain you are legally able to drink at the age of 18 so to celebrate you would go out clubbing for the first time or go to the pub and buy your first drink. That's more of a tradition. At the age of 18 your are also legally considered a adult, are able to vote in the general election and can buy cigarettes. I'm not too sure why the 21st is considered special since you can't really do anything new

I can't speak to the British, but in the United States the ages of 18 and 21 are important legal milestones. 21 once was the traditional "age of majority," the age of becoming a legal adult. It is very complicated and like many things in the U.S. varies a lot from state to state, but legally in the United States most aspects of adulthood begin at age 18. 18 is the age at which you can vote; the age at which you can sign legal contracts; the age at which you can be conscripted into the military; the age at which you can serve on a jury. If you commit a crime it is the age when you can be tried as an adult and be sent to an ordinary prison rather than a facility for "youthful offenders." In many states, however, 21 is the "drinking age," the age at which you can legally buy alcoholic beverages and be served at a bar.