The term "business cycle" is used to refer to the cycle in which economic activity in an economy rises and falls. It can be seen as the "ups and downs" of the economy.
There are four parts to a business cycle. There is a trough, which is the point when economic activity (as measured by real GDP) is at its lowest. After that comes recovery or expansion. Here, real GDP is rising. This continues until output hits a peak, which is its highest point. After hitting a peak, it goes into recession. This is when real GDP is declining.
So far, it appears that this sort of cycle is inevitable in an economy. As recent events have made clear, we do not know at this point how to end the business cycle and create a system which gives us permanent growth.