Comparing the Proton-Proton and Carbon Cycles
     The proton-proton cycle is a series of nuclear reactions which convert hydrogen nuclei (protons) into helium nuclei (alpha-particles). All Main Sequence stars do this, in one way or another. For stars like the Sun, the process almost always begins with the collision of two protons, hence its name, the "proton-proton" cycle. In massive stars, the process almost always involves carbon atoms catalyzing the reactions, and in the process being changed from carbon to nitrogen to oxygen, and at the end, back to carbon; hence its name, the "carbon" cycle, or using the abbreviations of the heavy atoms involved, the "CNO" cycle.
     Each "cycle" can be represented as a "chain" of reactions, proceeding one after the other. Sometimes, only one such chain is involved in a particular reaction, but usually, there are several ways in which particles can combine, which result in more or less the same end. In such a case, the chain which occurs most of the time (and usually produces the most overall energy) is the main chain, and those which occur less frequently are side chains.
(more to follow)

The Carbon Cycle
     In the Sun, the carbon cycle is of little importance (for reasons discussed below), but in massive stars (more than about 1.5 times the mass of the Sun), the carbon cycle is the normal way in which hydrogen is fused to form helium. And late in the Sun's life, when its core contracts and becomes substantially hotter than now, it will switch from proton-proton cycle to carbon cycle burning, so it is useful, even for the sun, to be aware of the basic principles of the cycle.

C + p —> N
N + p —> O
(various steps add protons, and have radioactive decays; see the text for more detail)
At t the end, there is a radioactive decay which produces Helium and Carbon (so you get the carbon back).

The carbon is not actually used up, but merely "catalyzes" the reaction. The actual reaction, as in the case of the proton-proton cycle used by the Sun, is:
4 H —> 1 He + 0.7% of the mass of the hydrogen —> energy

All nuclear reactions are sensitive to two factors: density and temperature.
Density is involved because it controls the rate of collisions. The higher the density, the more particles are running into more particles, so the number of collisions between the nuclear particles increases as the square of the density.

The higher the temperature, the faster the reactions go, as well, but the rate is very sensitive to the kind of reaction involved.
For MOST nuclear reactions of this sort, the temperature sensitivity is extreme: the 15th to the 35th power of the temperature.
For the proton-proton cycle, the temperature sensitivity is mild:
The 3rd to 4th power of the temperature (more sensitive at lower temperatures, less at higher).

This means, if you increase the temperature by 1%:
The proton-proton cycle goes 3 to 4% faster, but
The CNO cycle goes about 50% faster.

If you double the temperature
The proton-proton cycle goes about 10 to 20 times faster
The CNO cycles goes billions or hundreds of billions of times faster.

In the diagram below, the proton-proton cycle is shown as producing a little more energy as the temperature increases, while the carbon cycle is shown as producing a LOT more. At a certain temperature (shown by the intersection of the "curves"), the two reactions are equal; but at higher temperatures, although the proton-proton cycle produces even more energy than at lower temperatures, the rapid increase in the rate of carbon-cycle (hydrogen) burning is so great that for all practical purposes, it is the only thing of importance.
Conversely, at lower temperatures, such as in the core of the Sun (indicated by the vertical black line), the proton-proton cycle doesn't produce as much energy as at higher temperatures, but the carbon cycle produces so little energy, that it doesn't count at all. As a result, we say that lower-Main-Sequence stars, like the Sun, use the proton-proton cycle of hydrogen burning, and higher-Main-Sequence stars use the CNO or carbon cycle. But in both cases, the actual fuel is hydrogen, and the net result of "burning" the hydrogen is the same.