In 1905, from the special theory of relativity, Einstein showed that a tiny amount of mass could, in principle, be converted into a tremendous amount of energy: 
Following Einstein’s equation, it became clear that the Sun’s luminosity had to be explained by something completely different than everything imagined before. A process that would include the conversion of mass into energy. A process like “burning” but that would release much more energy per atom, and the speed of the “burning” would be much slower in order for the lifetime of the Sun to be consistent with the “known” age of the earth at that time based on evidence found in layers of rock.
In the late 19th century, from the knowledge of thermodynamics on heat transfer, most physicists had thought that heat was transported from the interior of the Sun to the exterior of the Sun by convection (transfer of energy by vibrations at a molecular level through a solid or fluid). But in 1894, R. A. Sampson suggested that the primary mechanism of heat transfer was radiation (the transfer of energy through electromagnetic waves). It will have to wait 30 years but this idea was eventually reused by Arthur Eddington.
In 1920, a lot of new discoveries were made. First, Eddington used the concept of radiative equilibrium to calculate the temperature at the center of the Sun and found it to be about 39 million K. Second, another scientist, Francis William Aston experimentally discovered by “accident”, since this was not the original goal of his experiment, a mass “deficit”. Using mass spectrometry, he made precise measurements of the masses of many different atoms, among them hydrogen and helium. Aston found that four hydrogen nuclei were heavier than a helium nucleus. This was not the principal goal of the experiments he performed, he was instead looking for isotopes of neon. Third, yet another scientist, Cecilia Payne, showed that hydrogen and helium were the most abundant elements in the stars (and our Sun).
In 1935, Eddington reduced his temperature estimate for the center of the Sun to 19 million K. However, Eddington’s first calculations back in 1920 made no assumption on how the Sun’s heat was produced: he proposed 2 alternative mechanisms: the mutual annihilation of protons and electrons OR the fusion of hydrogen atoms into helium atoms in some unknown manner (also called Eddington’s thermonuclear hypothesis).
Eddington’s thermonuclear hypothesis was directly inspired by Einstein’s equation and Aston’s discovery of a mass deficit. The importance of Aston’s measurements was immediately recognized by Eddington. The mass difference between hydrogen and helium meant that the sun could shine by converting hydrogen atoms to helium. This burning of hydrogen into helium would (according to Einstein’s equation) release about 0.7% of the mass equivalent of the energy. In principle, this could allow the sun to shine for about a 100 billion years.
The final piece of the puzzle and the first detailed explanations of the thermonuclear mechanism that Eddington hypothesized were published around 1938. There are actually 2 fusion reactions that were discovered: the proton-proton chain reaction and the CNO cycle (for carbon–nitrogen–oxygen).
The proton-proton chain reaction came from a group of scientists (Edward Teller, Charles Critchfield, Hans Bethe, George Gamow). They explained how energy could be produced at high temperatures by a chain reaction starting with proton-proton collisions and ending with the synthesis of helium nuclei. (see Charles Critchfield’s entry on Wikipedia)
The CNO cycle was independently proposed by Carl von Weizsäcker and Hans Bethe in 1938 and 1939, respectively.
Without going into details, the CNO cycle is the dominant source of energy in stars more massive than about 1.3 times the mass of the Sun, and the proton–proton chain is more important in stars the mass of the Sun or less (to be investigated further in another article).
Sources: Wikipedia and Encyclopedia of the Solar System. McFadden, Weissman, Johnson (2007)
Andromeda Geek 