On the most basic level, the carbon cycle explains how carbon is recycled on Earth, describing how the biosphere, hydrosphere, atmosphere and sediments exchange carbon. Below is an illustration showing how green plants absorb CO2 (think of it as their food), animals then eat the food and the carbon, and animals then release it back into the atmosphere. This is one of the more basic examples of a carbon cycle.
Organisms that photosynthesize (plants and phytoplankton) convert carbon to organic forms that are then consumed by animals and fungi. This process takes atmospheric CO2 and makes it available for others to consume.
Producers, consumers, and decomposing organism also give off CO2 through cellular respiration.
Below is a more complete picture of the carbon cycle, including the burning of fossil fuels, plant respiration, and the eruption of volcanoes, which all add CO2 to the atmosphere.
We generally think of four sources of carbon, also called carbon reservoirs: the atmosphere, the terrestrial biosphere (forests, non-living organic materials, freshwater systems, and etc), the oceans, and the sediments (fossil fuels). Carbon is exchanged between all four.
Check out this resource from NASA to learn more and read about North America's first State of the Carbon Cycle Report .
Sources: Campbell and Reece's Biology and Earth Observatory at Nasa.gov