In general, certain fusion reactions occur only when the conditions, like a high enough pressure or temperature, are present in the core of a star. In a star like the sun, for example, the predominant fusion reaction which occur are called the proton-proton chain, which has three main branches:
http://en.wikipedia.org/wiki/Proton–proton_chain_reaction
The reactions listed in the first branch, where protons fuse to form deuterium and then eventually helium, is the process which primarily occurs in the sun's core now. After helium forms, it essentially becomes inert, as far as fusion reactions are concerned. Helium gradually accumulates in the core of the sun, which currently is about 73.5% hydrogen and about 25% helium, with the rest being trace amounts of various elements up to and including iron. If the temperature gets hot enough, eventually branch two and three fusion reactions occur. In all of these reactions, elements like lithium, boron, and beryllium will form and then break down again constantly.