Red supergiant stars are the largest stars in the Universe by volume (meaning they also have the greatest diameter), however, they are not necessarily - and almost never are - the largest stars by mass.
Creating a Red Supergiant
Main sequence stars remain in hydrostatic equilibrium by converting the hydrogen in their cores into helium through nuclear fusion.
When low and medium mass stars deplete their hydrogen fuel their core begins to collapse, significantly raising the temperature - by nearly an order of magnitude. The energy escaping from the core during this process pushes the outer part of the star outward, forming a red giant.
The star will remain a red giant until the core reaches a high enough temperature to begin fusion helium into carbon and oxygen. At this time the star shrinks down slightly into a yellow giant.
A high mass star, however, will undergo a similar, yet different process. Because of their higher mass, when the core collapses after the hydrogen burning phase the rapidly increased temperature leads to the fusion of helium very quickly.
The energy released by the very high rate of helium fusion means that the star would not be stable as a yellow giant (like low and medium mass stars are at this phase) or even a red giant, but rather evolve directly into a red supergiant.
At this stage the gravitational force of the star is balanced by its immense outward radiation pressure caused by the intense helium fusion taking place in the core.
The process of evolving into a red supergiant, though, does come at a cost. They loose a considerable percentage of their mass. As a result, while red supergiants are counted as the largest stars in the Universe, they are not the most massive.
Properties of Red Supergiants
Red supergiants are red because of their low surface temperature, usually only about 3,500 - 4,500 kelvin. While their cores are extremely hot, the energy spreads out over the interior and surface of the star. According to Wien's law, the color which a star radiates most strongly is directly related to its surface temperature.
Because of their immense size and mass, these stars require an incredible amount of energy to sustain them and prevent gravitational collapse. As a result they burn through their nuclear fuel very quickly and live merely 10s of millions of years (depending on their actual mass).
Other Types of Supergiants
While red supergiants are the largest of the breed, there are other types of supergiant stars. In fact, it is common for high mass stars, once their fusion process passes beyond hydrogen, that they oscillate back and forth between different forms of supergiants. Specifically becoming yellow supergiants on their way to becoming blue supergiants and back again.
The most massive of supergiant stars are known as hypergiants. However, these stars have a very loose definition, they are usually just red (or sometimes blue) supergiant stars that are the highest order: the most massive and the largest.
The Death of a Red Supergiant Star
A very high mass star will oscillate between different supergiant stages as it fuses heavier and heavier elements in its core. Eventually, though, it will exhaust all its nuclear fuel that provides the exothermic reaction necessary to fend off the gravitational force that will bring about its collapse.
At this point the core is primarily iron (which has an endothermic nuclear reaction) and the core can no longer sustain sufficient outward radiation pressure and the core begins to collapse.
The subsequent cascade of events leads, eventually to a Type II supernova event. Left behind will be the core of the star, having been compressed due to the immense gravitational pressure into a neutron star; or in the cases of the most massive of stars, a black hole remains.