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Delegates represent a state's votes at a national party convention, in some proportion to the state's popular vote.
Each state gets a set number of delegates based on population. For example, California has 172 Republican delegates because a lot of people live there. Missouri, with a smaller population, has 52. In the Democratic Party, delegates represent votes proportionally (i.e. if Candidate A gets 60% of votes, 60% of the delegates will represent him or her). Republicans, however, usually use a "winner-take-all" strategy (Candidate B gets 60% of the vote, all the delegates will vote for him or her)
Any person eligible to vote can become a delegate, as long as he or she meets the state party's basic requirements, which should be publicized widely. In many states, you can be a delegate when you're 17 -- as long as you will turn 18 by national Election Day.
Most delegates are voted in at a local level to represent individual districts or counties. Each state determines its own process by which its delegates are selected.
Although every voter voices his or her choice through a ballot, delegates ultimately choose candidates at the national conventions. Often the popular vote is the same as the delegates' representation, but not always. For example, Hillary Clinton won the popular vote in Nevada's 2008 primary, but Barack Obama won more delegates than she did.
The system of delegates was devised at the first Constitutional Convention in 1787 to make sure all districts and communities are represented fairly. The delegate system forces candidates to gain support throughout the country, instead of one or two heavily populated areas of it.
National parties ultimately govern nominating rules and can take away states' delegates. In 2008, some states moved their primary and caucus dates up to January. The Democratic and Republican national parties penalized these states by taking away all or part of their delegates because they went outside the timeframe established by the party.