A recession is a period of time that lasts more than a few months where the economy gets significantly worse; a depression is defined as a severe recession where things plummet dramatically. A recession does not always lead to a depression, but a depression is always the result of a recession.
There is no standard definition for either a recession or a depression, but there are some commonly accepted facts on both economic perils. A recession is defined as a period of at least a few months where the real GDP is slowly declining. There may be a period within those months that the GDP rose briefly, but plummeted again. A recession can also be seen in the way that companies do business, in employment rates and in industrial production.
When a recession turns into something more severe, it is considered a depression. A depression usually starts with a gradual recession and truly begins when the GDP and economy takes a fast drop in the way it is operating. The most famous of these depressions in history was the United States in the 1930s. The economy was booming, but gradually began to fail. It took a steep drop very quickly when the stock market crashed.