This is meant to add to @BKay's answer. Here are some examples that might help to see what's happening when a bank makes a loan.
A bank's balance sheet
The image below is text taken from Greg Mankiw's textbook Macroeconomics (7th edition, p. 555). This is a simplified version of a typical bank's balance sheet. Notice that reserves, loans, and securities are considered assets and that deposits, debt, and equity are considered liabilities.
What happens when I deposit money into a bank?
Now, let's first consider what happens when someone deposits money into this bank. For example, suppose someone deposits \$10. The new money goes to the vault which may then be deposited in the bank's account at the Federal Reserve (the central bank). This is a new asset. At the same time, the new deposit is a new liability. All together, this will increase the reserves of the bank by \$10 and the deposits by \$10. Both the assets and liabilities of the bank increased by \$10.
What happens when a bank makes a loan?
Now, what happens when a bank makes a loan? Suppose a bank makes a loan to someone for \$10. In the picture, it's clear that this will increase the amount of loans from \$500 to \$510. Now, remember that loans are considered assets. That is, the borrower has an obligation to pay the bank back.
But, where will the money come from? A few things can happen. One example of what could happen is the following. The bank could use its reserves. In this case, reserves would drop from \$200 to \$190. However, in a fractional reserve system, banks have reserve requirements. The ratio of reserves to deposits must stay above a certain percentage. More realistically, loans might go up by \$11 (because of interest), reserves down by \$10, and owner's equity up by \$1.
(However, I believe that often times, a bank might issue the loan but immediately sell the new asset to another party, while taking only a small cut.)
The benefits of fractional reserve banking
Economists typically agree that the fractional reserve system adds some instability to the economy. However, it also adds many great benefits as well. There is a big literature about the trade-offs here and the many functions of (fractional reserve) banking, in general. (One of the most important economic functions is that it allows us to better smooth out liquidity shocks. See an important paper by Diamond and Dybvig).
There is simply too much to get into here. You might consider opening a new question that specifically asks for a list of the summarized benefits. Also, check out Mishkin's Money and Banking book (undergraduate level) or Microeconomics of Banking by Freixas and Rochet (graduate level, but still very accessible).