What is Money?
Money is a medium of exchange for goods or services within an economy.
How Money Works
Philosophically, anything can be money, but coins and paper notes are the most generally accepted forms. In most cases, each country in the world has its own money, but in many cases several countries use the same money (such as the Euro). A country's government designs and manufactures that country's money.
Some money is fiat money, meaning that it has no intrinsic value. That is, the paper or metal used to create the money is not worth very much in terms of its value as a raw material. Most paper money is fiat money, and its value comes from what it represents rather than what it is. Before 1971, the U.S. dollar was not fiat money -- it was backed by a corresponding amount of gold held with the Federal Reserve.
The foreign exchange markets are places to trade money, and these markets affect exchange rates (that is, the amounts of one money needed to buy a certain amount of another money).
Why Money Matters
Most money only has value because people want it. This idea is what made beaver pelts, shells, peppercorns, tulip bulbs, and other things into money at various points in history. However, when the demand (or fashion) faded for some of these goods (or more people found they really needed corn instead of beaver pelts), these systems became cumbersome. Paper money solves these problem because it is exchangeable for any good or service that people want (rather than just beaver pelts).
This isn't to say that paper and coins aren't the only forms of viable money today. Quite often, companies use shares of their own stock as money to acquire other companies, and anybody who has ever watched a crime show knows that cigarettes can buy a in prison.