There are several different types of fusion which can occur in stars at different points in their life cycle depending on their mass and metallicity ("metal" means something different to astronomers than it does to most people).

In general, and tremendously simplified, the most common net result of stellar fusion is that one starts with protons (hydrogen nuclei) and ends with alpha particles (helium-4 nuclei). However, there are also (lots of) other things going on, read on if you're interested in more details.

In our own Sun, the most common type of fusion is called the proton-proton chain reaction. In the most common branch of this process, two hydrogen atoms fuse to form deuterium, which fuses with another hydrogen to form helium-3, and then two helium-3 atoms fuse to form helium-4, releasing two hydrogen atoms in the process.

(Note: everywhere I've said "atoms" above I really mean "nuclei"; at the temperatures and pressures required, atoms don't actually exist as independent entities and it's more of a plasma of nuclei and free electrons).

Also, the last step in the process above is called the PPI branch and happens at temperatures of about ten to fourteen million kelvins. At 14-23 MK, the PPII branch is predominant, and involves a helium-3 nucleus fusing with a helium-4 nucleus to form beryllium-7, which then absorbs an electron to form lithium-7, which fuses with a proton (hydrogen nucleus) to form two helium-4 nuclei.

Above 23 MK, the PPIII branch dominates, which again involves fusion of helium-3 and helium-4 to make beryllium-7, which then fuses with a proton to form boron-8, which undergoes beta decay to beryllium-8, which is unstable and immediately splits into two helium-4 nuclei.

There's another branch which has been predicted but never directly observed (PPIV, the fusion of helium-3 with a proton to directly form helium-4). If this does occur in the Sun, it's only in very small amounts (less than about 0.3 parts per million).

In stars more than about 1.3 times the mass of the Sun (and to some extent in less massive stars, including the Sun itself) there's another process called the CNO cycle which has the same net result (protons are consumed and alpha particles are produced). I'm not going to go into the details, because this answer is already long enough, but if you're interested you could look up "CNO Cycle" on, say, Wikipedia.

Heavier nuclei (up to nickel-56) are built by the alpha process. Three alpha particles (helium-4 nuclei) can fuse to form carbon-12 (technically, two first fuse to form beryllium-8, but beryllium-8 is so unstable that unless it immediately fuses with another alpha particle it will just decay back into two helium-4 nuclei again). After that, the elements are built up by the addition of successive alpha particles:

carbon-12 + alpha -> oxygen-16

oxygen-16 + alpha -> neon-20

neon-20 + alpha -> magnesium-24

magnesium-24 + alpha -> silicon-28

silicon-28 + alpha -> sulfur-32

sulfur-32 + alpha -> argon-36

argon-36 + alpha -> calcium-40

calcium-40 + alpha -> titanium-44

titanium-44 + alpha -> chromium-48

chromium-48 + alpha -> iron-52

iron-52 + alpha -> nickel-56

Each of these requires higher and higher temperatures to sustain, so only the highest-mass stars reach this point. Above this the process falls apart, since zinc-60 would be the next product, but this fusion reaction is endergonic (absorbs energy rather than releasing it) and the core of the star collapses instead.

Heat and pressure from the collapse rebounds in a supernova event, which has plenty of energy floating around and can form all kinds of weird heavy nuclei. Supernova explosions are basically the source of all nuclei heavier than nickel.

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