What are business cycles?

Business cycles are short-run fluctuations in the economy's GDP, around the long-run trend rate of growth.
Over each business cycle, the economy will first go through a period of expansion, until it reaches a boom. At this point, unemployment will be low, as growth in the economy has led to more demand. This has created a need for more workers - which gives more opportunity for unemployed people to find job vacancies.  Here, there is also inflation, due to the excessive demand. During a boom, GDP rises far above the trend rate of growth.
Following this, there will be an economic downturn. Employment will fall as demand also drops, along with confidence. Unemployment will rise, and the economy may drop into a recession - where there are 2 consecutive quarters of negative economic growth. The economy contracts.
This cycle will carry on indefinitely, swinging from an upturn to a downturn and so on, around the trend growth line.