The four spheres of Earth are lithosphere (land), biosphere (living things), hydrosphere (water) and atmosphere (air). The processes by which the carbon flow from one to another, without increasing its total amount, are all part of the Carbon Cycle.
From atmosphere to biosphere: The carbon moves from the atmosphere to plants (biosphere) being attached to oxygen, forming carbon dioxide (CO2) used by plants in photosynthesis. When the plant is eaten by animals, the carbon starts flowing amongst the biosphere through food chains.
From biosphere to lithosphere: When living things (part of the biosphere) die, their bodies decompose and, through that process, the carbon in its body goes back to the soil (lithosphere). This can also happens in the water, when the carbon goes to the hydrosphere. Some of them can be burried and their decomposition may e blocked, so they become fossil fuels after millions of years.
From lithosphere to atmosphere: decomposers break down dead organisms, and in this process they return carbon as carbon dioxide to the atmosphere by respiration.
From biosphere to atmosphere: Animals (biosphere) can also release carbon to the atmosphere through respiration. We breathe in the oxygen and breath out carbon dioxide (which can be used by plants in photosynthesis).
From atmosphere to hydrosphere: The oceans and other bodies of water can absorb some carbon from the atmosphere, which is dissolved in the water.
There are several human activities that releases carbon dioxide to the atmosphere, but they won't be listed because they are harmful to Nature, increasing the amount of carbon dioxide available, therefore increasing the amount of carbon and causing a serious imbalance on the Carbon Cycle.