The concept of money has changed over the years but for our purposes, we can discuss the present-day functions of money. In fact, by definition, money is anything that fulfills certain primary functions in the market.
1. MEDIUM OF EXCHANGE: Money can be used for the payments of goods and services. This circumvents a lot of the problems that arise from a pure barter system.
2. UNIT OF ACCOUNT: In this context, money provides a way of measuring the value of goods and services in the marketplace. This also means that every unit of money is worth the same. My dollar bill is equal to anyone else's dollar bill, assuming none of them is counterfeit; this property of sameness is called fungibility.
3. STORE OF VALUE: Whatever money you make and have left after your expenditures can be reliably saved without losing value. Some people argue about this point because inflation depreciates the purchasing power of money.
4. STANDARD OF DEFERRED PAYMENT: Money is the accepted means of repaying debt from a loan.
Many forms of currency existed before paper money, or banknotes, first emerged (Tang Dynasty China, 7th century). People have used metals such as gold and silver but also shells, rocks, beads, and salt ("salary" comes from the Roman word for salt). The problem was that these other forms of exchange had problems with portability, one of the four primary monetary functions, or some combination.
Originally, paper money were banknotes which were IOUs from a bank. These were checks that could be traded in for metal coins at the bank. As nations became more advanced and unified, federal banks set the standard notes for a given country. Still, the paper notes derived their value from the promise of gold or silver.
What we use in the United States is called fiat money. When you hold that green, miniature portrait of Benjamin in your hand, you are holding a piece of paper that has been legally approved (legal tender) by the U.S. Federal Government as U.S. money. There are some nuances with this non-intuitive derivation of value but it boils down to trusting the federal government (its credit) since the bills themselves have no intrinsic value; in some sense, they're just paper. The important takeaway is that U.S. dollar bills are not convertible by law to anything else (like gold and silver) and their value isn't fixed. This can easily be witnessed by observing the fluctuating currency exchange rates between the dollar and the euro.
There are two things that significantly affect the value of the dollar, the supply of and demand for U.S. dollars. The Federal Reserve constantly prints out money to inject the economy with more cash which leads to inflation. Inflation is a purely monetary phenomenon where the prices of goods and services increase together. This is why the value of one dollar today is worth more than the value one dollar next year. Another way the dollar can lose value is if people (including foreigners) become less trustful of the American federal government. In this sense, there is a certain vulnerability to fiat money since it is entirely reliant on U.S. credit.
Now that we have some understanding of our monetary system, it should be obvious that a lot is at stake if the U.S. economy remains in deep debt to other countries and falls into a deeper recession. Simply printing out more money and selling bonds to other countries may provide short-term solutions but we need to come up with better solutions now for the long-run. Both parties disagree on tax raises and spending cuts which puts us in an uncomfortable situation. The only policy that both agree on is improving public education because that really is the best option for long-run economic growth, investing in human capital for a more productive workforce.