In nature, there's a constant flow of matter -- all kinds of materials -- from the earth, the oceans, the atmosphere and the living creatures, in all directions. This is the first time I hear about a combined Carbon-Oxygen cycle, I learned about the Carbon cycle as a separate one, but I guess it makes sense to talk about the two of them together.
Basically there are four processes in this cycle:
1. Photosynthesis, when plants take up carbon dioxide from the atmosphere and water from the ground and create sugar (which stays inside the plant) and oxygen (which is emitted to the atmosphere).
2. respiration - animals breathe oxygen from the atmosphere, use it to create energy, and emit carbon dioxide to the atmosphere.
3. combustion - most of it happens when people burn fossil fuels, which releases a lot of carbon dioxide to the atmosphere.
4. decomposition - when animals and plants die their bodies are decomposed and CO2 is released to the atmosphere.
Basically this cycle was in a steady state for millions of years, meaning that the rate of CO2 emitted to the atmosphere was equal to the rate of CO2 taken from the atmosphere, so the total amount of CO2 in the atmosphere remained constant. Since the industrial revolution, the rare of burning fossil fuels has increased by a huge amount, and the processes of taking up CO2 from the atmosphere can't keep up with the increased rate of burning fuels, so the CO2 concentrations in the atmosphere have doubled over the last 200 years or so.