A red dwarf is a main sequence star of small size (between 0.1 and 0.4 Sol masses) and low temperature (spectral class M). Since the luminosity of a main-sequence star is roughly proportional to mass^4, a red dwarf is typically around 10,000 times dimmer than Sol. However, a red dwarf also lives an extraordinarily long life compared to Sol: a star's lifetime on the main sequence is roughly proportional with 1/mass^3, so a red dwarf will typically last around 1,000 times longer than Sol. Compared to all the other stars around, red dwarfs are immortal.
Unlike a G-class yellow dwarf like Sol, when a red dwarf dies it doesn't enter the red giant stage; only a star big enough to fuse helium can become a red giant. Instead, it merely contracts into a white dwarf and slowly fades to a black dwarf over the next trillion years. Of course, no red dwarf in the entire history of the universe has died yet. They won't start dying off until you start talking about the timescales of the Stellar Era.
Some famous red dwarf stars include Proxima Centauri, Barnard's Star, and Wolf 359. Red dwarf stars are estimated to outnumber all larger stars combined by at least 2:1 (and probably closer to 9:1). However, the fact that they are very faint prevents us from detecting them at distances greater than 15 LY, a mere stone's throw compared to the size of the Milky Way.