A capital city is the main administrative and governmental centre of a nation. It is generally home to the head of state (the monarch or president), to the parliament or general assembly, and the the highest courts of the land. Most world capitals comply with these rules; a notable exception is in the Netherlands, whose capital, Amsterdam, is the home of the monarch, but the home of government is The Hague.
The capital need not be the largest city in a nation. The United States capital, Washington D.C., is far from being the largest city, and the same applies to the canadian capital Ottawa, the Australian capital Canberra and a host of others. Especially in new nations like the US and Australia, a new capital is established, or a small, uncontentious city chosen, to defuse rivalry between other larger cities.
As well as dominating politically, capital cities can often (although not invariably) dominate in commercial and cultural terms as well. Cities such as London and Paris have a political, economic and cultural hegemony. However, while Berlin is the capital of Germany, the main German financial centre is Frankfurt, and that of the United States is New York City, which also, with Los Angeles, is one of the nation's two 'media' capitals.