Stellar Structure:
 At its most basic, the life of a star is a struggle between gravity (which tries to crush it) and pressure (which can hold it up). For main sequence stars, the pressure is generated by heat, which is generated by nuclear fusion (in particular, hydrogen burning). High density and temperature are needed to allow fusion, so the star collapses until its core reaches a state where enough fusion takes place to halt further collapse. The conditions when this occurs vary by stellar mass (more massive stars must generate more pressure to hold up their greater weight); this is the reason for the "main sequence" which just expresses (with observable quantities) the amount of fusion and the size the star takes.

Of course, the heat generated by fusion does not stay in the core. It is passed out through the star either by direct radiation (carried by photons) or by convection (carried by motion of hot gases). The choice between these is made by whichever is easier - radiation is preferred unless the material is too opaque (non-transparent) or the amount of energy to be carried is too great. Cooler material is more opaque, so the outsides of cool stars are convective and their interiors are radiative. In hot stars so much energy is generated in a small volume in the core that the core is convective, while the larger envelope stays hot enough that radiation is preferred. Energy always leaves the surface of the star by radiation - that is why stars shine.

Nuclear Terminology:
Nuclei consist of protons and neutrons. An "element" is defined by the number of protons in a nucleus. The various numbers of neutrons that can go with a certain number of protons yield the "isotopes" of that element. The mutual repulsion of the protons is overcome by the "strong nuclear force". This force works like velcro: the protons have "hooks" and the neutrons have "loops". So protons can't stick to protons, and neutrons can't stick to neutrons. This means that there should be approximately the same number of neutrons and protons (to provide sufficient "glue"). One proton by itself is also called "hydrogen". A bare helium-4 nucleus (2 protons, 2 neutrons) is sometimes called an "alpha particle". Neutrons are not stable by themselves; outside nuclei they will decay in about 10 minutes to a proton and an electron. This is mediated by the "weak nuclear force". One can also have a proton turning into a neutron under special circumstances; it then dumps its positive charge as an anti-electon (positron) and a neutrino (a very low mass, very weakly interacting particle that carries off other needed quantum properties). Neutrinos are so weakly interacting that they can pass right through the whole Sun (on average) without hitting anything. The positron is an example of antimatter: each kind of particle has a corresponding antiparticle; when the two meet they are fully annihilated yielding the energy given by E=mc².

Stellar Fusion and the production of elements:

The main sequence phase in a stars life is the time during which it is converting hydrogen to helium in its core. In stars up to about twice the mass of the Sun, hydrogen burning takes place by the "p-p chain" (proton-proton). Any fusion requires that the mutual electric repulsion of the protons in a nucleus be overcome by slamming them together sufficiently hard that they approach close enough that the "strong nuclear force" makes them stick. The higher the temperature, the faster they move; the higher the density the more collisions there are.

The rules of the strong force mean that protons cannot fuse by themselves (in addition to overcoming their repulsion we also need to have neutrons around). And free neutrons are basically not there, because they are unstable. So the pp chain starts by colliding 2 protons together and having one of them change to a neutron at the same time. This is very unlikely -  it takes the average solar proton ten billion years for it to happen (which sets the lifetime of the Sun). But there are a lot of collisions taking place in a stellar core! A positron and neutrino are emitted; the positron immediately annihilates an electron and energy is produced. The fusion makes deuterium (p+n), and it is then relatively easy to add another proton, making "helium-3" (p+n+p). The mass of this is slightly less than of the particles that made it up (binding energy is lower) and some energy is emitted. Finally, 2 helium-3 nuclei can collide and yield an alpha particle or helium-4 nucleus (2p+2n) plus two fast moving protons (and more energy).
The net result is that 4 protons have been made into an alpha particle, and 0.7% of the mass energy of the particles has been converted to heat. The neutrinos that are produced escape immediately, providing us with an instantaneous probe of energy production in the Sun. It was worrisome that only a third of the expected neutrinos are seen; this is called the "solar neutrino problem". We now think that this is because there are 3 types of neutrinos, they can change types, and we are only measuring one of the types.
In more massive stars the hydrogen burning proceeds in a different way, using carbon as a catalyst. That means that the carbon is left unchanged in the end, and 4 protons are converted to helium (using carbon, nitrogen, and oxygen as intermediaries). This process (which is also hydrogen burning) is called the "CNO cycle". The CNO must have already been there; these are the most common products of stellar fusion and are the most abundant elements after H and He. Carbon is produced in stellar cores after hydrogen is burned to helium.

When the temperature and density are sufficiently increased by further core collapse, 3 alpha particles are combined together (through intermediate steps) to make a carbon nucleus. This is called the "triple-alpha reaction", and goes on in stars when they are giants. Low mass stars must stop here, but high mass stars can go on to make all the other elements. The stars that are forming today contain elements that were produced in previous generations of stars. As, of course, were all the elements in your body (except hydrogen).

Here's another discussion of stellar energy cycles.