Hypothetical and unlike red dwarfs, brown dwarfs, and white dwarfs, a black dwarf is a stellar remnant. It's specifically a white dwarf that has cooled sufficiently that it no longer emits significant heat or light. The time required for a white dwarf to reach this state is calculated to be longer than the current age of the universe, which is estimated to be 13.8 billion years.
Therefore, black dwarfs are expected to exist in the universe presently and the temperature of the coolest white dwarfs is one observational limit on the age of the universe.
The name "black dwarf" has also been used to describe sub stellar objects that do not have sufficient mass, less than approximately 0.08 solar mass to maintain hydrogen-burning nuclear fusion. Said objects are now generally called brown dwarfs, a term coined in the 1970s. Black dwarfs should not be mistaken with black holes, black stars, or neutron stars.
Since the hypothetical black dwarf is just a white dwarf that has cooled completely, it should be the same structure as a white dwarf. The final end product of fusion is iron. Ergo, a black dwarf would be made of iron. The iron is exremely dense since gravity would have squished it all together in one mass of subatomic particles, but it's iron regardless.