There are two carbon cycles, the biological one (fast) and the geological one (slow). In the biological cycle carbon enters food chains through autotrophs (self-feeders, nearly all of them photosynthesising organisms like algae and plants) which capture carbon dioxide from air or bicarbonate ions from water and use them to make organic compounds such as glucose. These in turn get eaten by heterotrophs (other-feeders) such as animals who consume the organic molecules and break them down through cellular respiration where carbon in the molecules is released as carbon dioxide. Decomposers (fungi and bacteria) also release organic compounds and carbon dioxide when they break down dead organisms and waste products. Rinse, repeat.