Fractional-reserve banking is a form of banking in which banks are only required to keep a fraction of their total deposits on hand. Most banks around the world use this system, because fractional-reserve banking is what allows banks to generate funds. It's also what allows people to receive loans from banks, or open interest-generating accounts. The alternative to fractional-reserve banking is full-reserve banking, in which a bank must be able to hold all of its deposits on hand.
Some form of fractional-reserve banking has been practiced for a very long time in the banking industry. The way fractional-reserve banking works is that the bank essentially borrows from its depositors to offer loans to people who apply for them. Banks may also choose to invest deposited funds in various ways. If you bank in an institution which uses the fractional-reserve system, this means that you are indirectly funding the loans and investments made by the bank; so if you bank at the same institution which administers your mortgage, you could say that you loaned yourself some of the money!
The advantage to fractional-reserve banking is that it allows banks to generate income on the funds deposited. Every time your bank borrows from you to make a loan to another bank customer, it gets to charge interest on the loan, pocketing the interest. If you have money in an account which generates interest, you get a cut of the interest charged on loans, but the bank still pockets a significant portion of it. Fractional-reserve banking is big money in a very literal way, which is why so many banks like this system.
The disadvantage of fractional-reserve banking is that it puts banks in an awkward position when it comes to liquidity. While banks aren't required to retain their deposits on hand, they have to be able to redeem deposits upon request, as for example when a customer goes in to close a checking account. If a group of depositors all ask for their money bank at once, in a situation known as a bank run, the bank may not have enough funds on hand, which could be a serious problem.
Liquidity problems can be compounded when a bank makes poor lending decisions, and borrowers default on loans. When a customer defaults, the bank loses the borrowed money, along with the income from interest, and it must scramble to make up the shortfall. Too many bad loans can cripple a bank, causing it to become insolvent.
Many nations support fractional-reserve banking, with agencies like the Federal Reserve Bank in the United States acting to regulate and oversee fractional-reserve banking. To address depositor concerns, some nations have government agencies which insure deposits up to a certain amount, and these agencies may also perform regular audits on the banks which they back to ensure that they are not taken by surprise when a bank becomes insolvent.