Toxic black mold is a name given to a type of black mold technically known as Stachybotrys chartarum. According to The Money Pit's indoor air quality expert, Jeff May, this type of mold is often referred to as "toxic black mold" because it's black, and under certain conditions, this type of microfungus, along with others, such as species of Aspergillus and Fusarium molds, can produce mycotoxins which have an impact on human health.
Other kinds of molds can be black, too, including Cladosporium mold, the most common mold, and "toxic black molds" produce mycotoxins only under certain conditions, and not all the time.
Stachybotrys chartarum is not as much as a concerns and another species of Aspergillus mold, for several reasons. Stachybotrys chartarum requires wet conditions (steady leaks or a floor flood) in order to thrive, and thus is not as common as Aspergillus molds, which can grow in conditions of elevated relative humidity. And Aspergillus spores are smaller than Stachybotrys spores, and thus can remain airborne longer and easier to breath.
Worrying excessively about toxic black mold also takes the focus away from what is considered a greater threat: mold allergens that, in general, can affect those who are sensitized.