“Dwarf” was originally a term used to distinguish between the two kinds of red stars in the universe - very massive, and very small. These were termed “red giants” and “red dwarfs”. The dwarf terminology gradually expanded to mean “not giant” stars of any colour, and the line between “giant” and “dwarf is somewhat poorly defined; the Sun is technically a "yellow dwarf” star.
What most people think of when they hear “dwarf star” are brown dwarf, red dwarf, and white dwarf stars. There are also a few theoretical kinds of dwarf stars, which is where black dwarfs fall. These stars are all classified based upon their colour, although confusingly these are not usually the colors they would appear to our eyes. (Brown dwarfs, for instance, would appear a deep pink - see above for 3 brown dwarfs as they would appear to us.)
Yellow and red dwarf stars are normal stars - they burn hydrogen in their cores and live on the main sequence of stellar lifetimes. Red dwarfs are smaller than our sun, only getting up to 50% the size of our sun. As a result, their surfaces are cooler, hence the colour shift towards the red. They don’t consume their hydrogen as quickly as our sun does, so even though they’re less massive and thus have less hydrogen, they still live for a much longer time than our sun will. Because red dwarfs require less matter to create, they are the easiest to make. Red dwarfs are therefore the most abundant type of star in the galaxy - our nearest stellar neighbor is a red dwarf.
Brown dwarfs are failed stars. They’re essentially massive Jupiters - large collections of gas that are not massive enough to create the pressures required to start burning hydrogen into helium. These dwarfs can be pretty cold; there was one found not too long ago that was only as warm as a cup of coffee. A brown dwarf can’t do anything except sit there and slowly radiate away its heat - it won’t ever become a fully fledged star. The iron rain you refer to was the conclusion of a study from 2006; evidence was found that at the temperatures of the star they were looking at, the iron they detected in its atmosphere should be forming liquid droplets and raining down towards the surface of the star. Further studies have found evidence for massive, Jupiter-style storms in the atmospheres of these stars. The behavior of the metals and other elements in a brown dwarf’s atmosphere will depend strongly on the temperature of the star in question. Since “brown dwarf” is a rather broad term, some of these stars will be too cold for iron rain, and some will be too warm. Of course, the presence or absence of a particular element will depend on the gas the dwarf formed out of, since the brown dwarf is not building any new elements itself.
White dwarfs are the most exciting to make. They are what is left over after a main sequence star (like our Sun) dies. The star will have gone through the red giant phase, and then shrugs off its less dense outer layers into a planetary nebula. At the end, all that is left is a hot, dense core of what was once the centre of the star in a volume about that of the Earth. They are so dense that the pressure provided by the electrons of the atoms within the star pushing against each other is what keeps them from becoming any smaller, and so hot they glow white just from trapped heat. This is the end-point of our Sun.
The black dwarf - still a theoretical object - is the name we would give to a white dwarf star which had managed to completely lose all of its heat, effectively going completely out. The length of time it takes for a white dwarf star to lose all of its heat is longer than the length of time the Universe has been around, so we don’t expect to see many of these around.