The method below is not the only way to make carbon. Helium nuclei can have just about any number of neutrons - generally two or three - but for this entry we shall simply assume they all have two. It can also be made by radioactive decay of other large atoms, or in the case of Beta decay, smaller atoms.
This entry deals specifically with the type of fusion reactions that may take place inside giant red stars.
Carbon is created in the heart of giant red stars. Three alpha particles, a helium nucleus, two protons and two neutrons collide to give a nucleus with six protons and six neutrons, carbon-12 nucleus. This can only happen in the middle of the stars where everything is plasma1 so the electron clouds do not make it harder.
Unfortunately the odds of such a collision happening, all three at one time, are very small. If all three had to collide at once, the number of collisions would not have made nearly as much carbon as we have now. So therefore, something else must be happening.
The collisions take place in two steps.
Two helium nuclei collide at the same time and create a beryllium nucleus. The only problem with step one is the form of beryllium given only remains for 1*10-16 seconds. This means that the third and final helium nucleus better get in quick.
Another helium nucleus then collides with the beryllium and makes a carbon isotope 6-Carbon-12 - six protons and six neutrons.
From the looks of things, that way of making carbon is another dead end; too many beryllium atoms would fall apart.
The energies of beryllium and helium are similar to that of carbon. When the energies are equal it creates a resonance. This near-resonance would make it easier to create carbon.
In the 1950s, Fred Hoyle said that carbon must have a resonant state, or else it would never be able to exist in the quantities we know of. By means of some nifty calculations (far too complex to go into here) he determined that it was 7.6549 MeV2. Unfortunately, the combined energies of beryllium and helium are four per cent higher than it. However, somehow this discrepancy is what we want.
The reaction takes place and carbon is created; strangely the mass of the carbon created is four per cent less than the combined mass the helium and beryllium. The carbon is said to be excited. As this is unstable, it emits the missing four per cent as gamma rays, balancing E=Mc2. These are then absorbed by the surrounding matter, giving out heat.
Note on Fusion
The reaction which joins two or more nuclei together is called fusion. Such a reaction takes massive amounts of energy to proceed and efforts to perform it have so far been found unsuccessful on Earth3. When done though, the formula E=Mc2 comes into play and energy is given off. This energy is part of what keeps the star hot and giving off light.