From Mike Moffatt,
Your Guide to Economics.
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A. Parkin and Bade's text "Economics" gives the following definition of the business cycle:

The business cycle is the periodic but irregular up-and-down movements in economic activity, measured by fluctuations in real GDP and other macroeconomic variables.

If you're looking for information on how various economic indicators and their relationship to the business cycle, please see A Beginner's Guide to Economic Indicators. Parkin and Bade go on to explain:

A business cycle is not a regular, predictable, or repeating phenomenon like the swing of the pendulum of a clock. Its timing is random and, to a large degress, unpredictable. A business cycle is identified as a sequence of four phases:
• Contraction (A slowdown in the pace of economic activity)

• Trough (The lower turning point of a business cycle, where a contraction turns into an expansion)

• Expansion (A speedup in the pace of economic activity)

• Peak (The upper turning of a business cycle)

A recession occurs if a contraction is severe enough... A deep trough is called a slump or a depression.

The difference between a recession and a depression, which is not well-understood by non-economists, is explained in the article "Recession? Depression? What's the difference?". The following articles are also useful for understanding the business cycle, and why recessions happen:

• Why Don't Prices Decline During A Recession?
• Do Changes in Stock Prices Cause Recessions?
• Are Recessions Good For the Economy?
• A Beginner's Guide to Economic Indicators

The Library of Economics and Liberty has an excellent piece on business cycles aimed at an advanced audience.