Fusion is the energy process in the Sun. Fusion means "putting together" to form something different. In this case it is hydrogen nuclei, which are simply protons, coming together to form a somewhat more complex helium nucleus. The specific process in the Sun is called the Proton-proton chain because it involves a series of interactions in which protons build up the helium nucleus, leaving additional leftover protons that can start the process over again, hence a "chain."
This process is sometimes referred to as thermonuclear fusion. "Thermo" because it occurs only at very high temperatures, and "nuclear" because unlike everyday energy processes such as burning gasoline, it directly involves only the nuclei of atoms.
In order for positively charged protons to come together to form a helium nucleus, they must overcome their natural electrical repulsion of each other. This repulsion is known as the Coulomb Barrier. The only way this can happen is if the protons are packed tightly together and are moving very rapidly. Basically this means that they must be in a highly compressed and extremely hot gas. In fact, at such high pressures and temperatures, it is too hot and energetic for electrons to stay connected to the protons to form normal atoms. Instead, there is a mixture of free electrons and protons in a kind of particle soup. This kind of particle (rather than molecular) gas is called a plasma and is the situation in the core of the Sun. Here the protons are moving so fast that -- aided by properties of quantum physics -- they sometimes slam into each other with enough force that they overcome the Coulomb Barrier and touch. At that point, nuclear forces come into play to "glue" the protons together in spite of their electrical repulsion. Through a series of such events, attended by the creation of exotic particles such as the neutrino and positron, a helium nucleus with two protons and two neutrons is produced. Two additional protons are left over and can continue the chain.
Physicists have measured the mass of a helium nucleus such as is produced in the core of the Sun. It turns out that it is a little less than 4 times the mass of the 4 protons that created it. This "mass deficit" is partly due to the positrons and neutrinos produced in the process. But even considering that, there is still a little of the mass that is missing. That mass was converted into energy by Einstein's famous equation, E=MC^2 ("Energy equals mass times the speed of light squared."). This is how the Sun produces energy.
At the Sun's core roughly 600 million tons of hydrogen (protons) is converted into about 596 million tons of helium each second. The mass deficit of about 4 or 5 million tons is converted directly into energy as per the paragraph above.
All of the Sun's energy is produced in this manner at the core of the Sun. The energy is vibrational (heat) as well as in the form of high energy gamma rays. The energy from the gamma rays slowly works its way out from the core of the Sun, changing many, many times into longer wavelength radiation. By they time it reaches the surface, it consists mostly of visible light photons.
[We should point out that because it involves only atomic nuclei, fusion is fundamentally different from any process normally encountered on Earth. Ordinary chemical reactions do not involve nuclei, but rather the electrons in the outer shells of atoms. Although the production of energy in "ordinary" situations do not involve the nuclei (and so do not change the elemental nature of matter), they still operate via Einstein's equation and there still is a tiny loss of mass when any energy is produced. However, it is much less than in nuclear reactions and is not normally even measurable.]