How many depressions have there been?
Prior to modern economic practices, depressions weren't uncommon. There are very obvious examples of depressions reaching back into the 17th century in Europe, though they weren't necessarily called economic depressions at the time. In America, we've only had one depression, and it was great.
The Great Depression lasted from 1929 to 1939, triggered by a crash in the stock market that sent investors running from the markets and regular people scrambling to pull their life savings out of banks. They pulled so much money out of banks that they caused many to fail entirely in the days before the creation of the Federal Deposit Insurance Corporation (FDIC), established as a direct result of those early days of the Great Depression.
It's unlikely that another depression will occur in the United States soon, however, due to a variety of controls that have been put in place. These include the ability of Congress and the president to enact fiscal policy that injects money into the economy either directly or through job creation, the Federal Reserve's ability to adjust monetary policy to encourage or discourage lending and borrowing, and fiscal policy measures to curb government spending when revenues are lean.