What is it?
Paper money is just that: currency printed on paper, tradeable for goods and services. It can be backed by ('pegged to') a real asset such as precious metals, real estate, or other commodities. More often in modern times, however, it 'floats' - meaning that it is not guaranteed to be convertible to any particular hard specie. Such currency is called 'fiat money,' as it has value only by fiat of the issuing body. Sometimes it is pegged to another nation's currency; this is still 'fiat money' if the other nation's currency is not convertible..
The U.S. and Britain both had paper money before officially pegging its value to gold. There are various types of economic pressure which push for paper money to be issued. First of all, pegging the value of paper money to a convertible asset means that ones' total economy size is limited by the availability of that asset - in other words, no matter how much business you're doing, if you can't mine enough gold to back your banknotes, your economy can't expand. This is a good thing if you're fighting inflation(I believe, but I'm not an economist; I expect to be corrected if I'm wrong). However, constrictions on gold supply (or silver, or whatever) mean that there is an external limiting factor on economic expansion.
One event that usually causes a huge demand for paper money is a war. Wartime spending is typically done with much less of a regard for the actual carrying capacity of a nation's economy; as a result, some degree of inflation is to be expected. In order for this spending to occur, a cheap form of exchange which does not rely on a scarce resource needs to be available - paper money, cheaply and quickly made, is a popular answer.
History and Technology
According to Nova, paper money first appeared in China around 806 A.D.. It was used for many years before vanishing again in favor of coinage, and it would be many more years before it would be reintroduced in the West. Early currency was not issued by governments; in fact, another name for it is 'banknotes', which indicates that early paper currency consisted of drafts on particular banks. Trust in those banks on the part of the users of such currency allowed for transactions to occur at a physical remove from the bank in question, without having to transport valuable (and bulky) coin. The shift from a letter of credit or draft to true money occurred when these banknotes became payable to the bearer rather than being bound to particular transactors.
American Paper Money
The first paper money in the Americas was issued by the Massachusetts Bay Colony in 1690. As part of its revolutionary agenda, the Continental Congress issued its own money in 1775 in a bid to solidify fiscal independance from England. Soon after the American Revolution, in 1791, the first Bank of the United States was established and granted authority to issue money by Congress. The Federal Reserve Act of 1913 laid out the system by which money in the United States was to be regulated, and provided for the first standardization of American paper money - the Federal Reserve Note, whose descendants remain in use today. Look at your nearest greenback; it will say "Federal Reserve Note" on it somewhere.
One significant barrier to paper money's acceptance is its fragility. People aren't going to want to trust their earnings to an exchange medium which is easily torn, faded, crumpled or otherwise rendered unusable. As a result, paper money is usually made not from wood pulp paper but from rag. The cloth fibers give the bills much greater flexibility and endurance; the average lifespan of U.S. Currency, according to the U.S. Mint, is as follows. Singles (one-dollar bills) last in circulation for around eighteen months. Fives last about fifteen months. Twenties remain in circulation for two years. Interestingly, tens have about the same lifespan as singles do. Larger denomination bills ($50, $100) can last up to eight years.
As long as those times are, however, they pale before the lifespan of a metal coin. This is offset, of course, by the fact that paper money is much, much cheaper to make. Despite its use of rag, it's still much more easily destroyed by chance; as a result, most issuing bodies have policies whereby damaged money will be replaced free of charge so long as the bills can be positively and uniquely identified. Did you know that? If you accidentally send your hoard through a shredder, there are people whose job it is to piece together and identify the bills, so that they can be officially removed from circulation and replaced. Of course, shredding them deliberately is a felony, but...well.
Of course, with the ease of manufacturing comes the danger of counterfeiting. After all, if a government can print money, why can't some enterprising entrepreneurs? Well, there's an entire industry making sure they can't. Embedded security devices, microfine patterns that are hard to duplicate, designs that produce moire patterns when electronically scanned, special papers and inks, holograms, serial number codes, the works.
As Douglas Adams noted so trenchantly, many humans seem obsessed with the movement of little pieces of paper, often to the exclusion of all else (even digital watches!). Capitalists think this is fine. Others don't. One thing that's curious, though - without what amounts to a collective agreement among all parties, paper money can't function. It is only exchangeable for goods and services if the seller is willing to take it; it is only worth working for if it can be used. There is enough historical 'inertia' behind the entire idea of paper money to keep the system working. Economic pressures and events reflect on the money itself; it takes serious dislocation (hyperinflation, local disaster) to depress confidence in the bodies backing paper money enough these days to make it truly worthless. When paper money becomes worthless, though, the self-supported system has collapsed - and that's what it's called.