Cladosporium is a very common mold. It can appear green, brown, grey or black on surfaces. It is classified as either Hazard Class B or C, depending on the species - i.e., allergic reactions are generally the only concern with Cladosporium. It can grow in many places, including walls, wood, dust, and insulation.
Aspergillus is another common mold. It can look grey, brown, yellow, green, white, or black. There are many species in the genus, and they generally fall into Hazard Classes A or B. Some can cause infection in people with weak immune systems, and some of them can make toxins in certain circumstances. Others will only cause allergic reactions. Aspergillus can grow on walls, insulation, paper products, soil, clothing, and many other places.
Penicillium is a name that often strikes people as familiar, and that is because modern antibiotics were discovered thanks to a species of Penicillium long ago. However, that doesn't mean that the genus can't be hazardous. It can look blue, green, or white, and its species are generally classified as B or C. It can be found on foods, such as cheese and fruit, or in the walls, the insulation, and other places.
Ulocladium usually looks black or grey. It falls into hazard classes B and C, and tends to grow in damp areas. It can be found on walls, around windows, in dusty areas, and other places.
Acremonium is often found on insulation and drywall/sheetrock, although it can be found in many other areas, too - it grows in damp places. It can appear white, grey or brown, and various species are found in all three hazard classes.
Stachybotrys is a harmful black mold. Stachybotrys is the infamous black mold that made the news in association with ill health effects many years ago. It needs a very damp area to grow, and is considered a Hazard Class A mold, as it can create toxins. It looks black on surfaces.
Alternaria looks similar to Ulocladium under a microscope. It can appear black or grey on surfaces. It is classified as a Hazard Class B mold, and has been known to cause various allergic reactions. It can grow on walls, dusty areas, around windows, damp areas, in soil, on plants, and in various other places.