Along with the terrestrial carbon cycle, there is another process labeled the "Carbon Cycle". This is a process of stellar nucleosynthesis where carbon nuclei serve as a catalyst for the formation of helium out of hydrogen.
All main sequence stars create energy by fusing hydrogen into helium. In stars the size of our sun and smaller, they do this directly: protons collide, and eventually form helium-4 from the p-p chain. In larger stars, the carbon cycle becomes more predominant. In this cycle, a stable carbon-12 nucleus absorbs protons, cycling through isotopes of carbon, nitrogen and oxygen, until the growing nucleus divides into the original carbon nucleus and a helium-4 nucleus. The cycle then repeats.
There are three salient points to remember about the stellar carbon cycle. First, it is strictly catalytic. While the proportion of carbon, nitrogen and oxygen will change, the total amount of these three elements will not. (Very occasionally, some slight amounts of fluorine and neon will be created, but only in negligible amounts). The second important point is that the reaction is very temperature and pressure dependent. In a star the size of our sun, the carbon cycle plays a minimal role in fusion (about 1%), but it rapidly increases with size, and in a star the size of Sirius, twice the size of our sun, it is by far the predominant producer of energy. The third thing about the carbon cycle is that it (surprise!) requires the presence of carbon within the star. Our sun is about 1/300th carbon, which is a relatively high proportion. In a star where this proportion was much smaller, the carbon cycle would not contribute to energy generation, even if the size of the star is quite large.
Although seemingly a technical detail, the existence of the carbon cycle has large implications for chemistry, physics, cosmology and astronomy.