General Astronomy/The Life of Low Mass Stars
The lifetimes of stars are determined by their mass, or the total amount of matter they contain. If an object is not sufficiently massive (at least about one tenth the mass of our Sun), it will not have enough gravity to squeeze itself tightly enough to heat itself up to stellar levels. Such an object will become either a planet-like body or an intermediate object called a "brown dwarf." A "brown dwarf" simply never initiates stable hydrogen fusion and by definition never becomes a full-fledged star. Some brown dwarfs go through a phase of limited "deuterium fusion", but this does not qualify to make them a full star. They eventually -- after very long periods of time -- simply go out and cool to the ambient temperature of the Universe.
Gas clouds (or just localized areas of larger gas clouds) that have at least one tenth the mass of the Sun, can get hot enough as they contract to start fusing hydrogen into helium. This is known a "stable hydrogen fusion" and it marks the birth of a true star. This requires temperatures in the millions of degrees. However, stars of such relatively low mass can sustain only very low levels of fusion, which is to say that they use their nuclear fuel slowly. So they do not grow very bright, and form small dim stars called red dwarfs.
Generally, red dwarfs go through their lifetimes uneventfully, just slowly "burning" their nuclear fuel in a stable fashion. Eventually, they will run out of nuclear fuel cool, and slowly become a burned out cinder known as a "black dwarf."
Do not confuse the terms brown dwarf, red dwarf and black dwarf. A brown dwarf has a mass too low to ever become a star. It produces energy, perhaps even by a limited type of fusion, but never enough for "stable hydrogen fusion. A red dwarf has low mass, but high enough to start stable hydrogen fusion, which qualifies it as a "real" star. However, it burns that fuel so slowly that it has a very long lifetime. It is likely that any red dwarf ever born since the Big Bang is still around, and has not yet turned into its final version, the "black dwarf." Considering the extremely long lifetimes of red dwarfs, it is likely that no black dwarfs have yet formed in the Universe.
Do not confuse the term "black dwarf" with "black hole", which is an utterly different object with a different origin.