Types of household mold
Grouped by what you should do rather than by taxonomy, because that is the question being asked. Note that three entries here are not fungi at all.
Not mold
Mistaken for mold constantly. Ruling these out is often the whole answer, and none of them needs remediation.
Worth investigating
Either a signal of a serious moisture problem, or a known allergen worth removing promptly. Not a reason to panic.
Stachybotrys chartarum
This is the species behind the phrase 'toxic black mold', and the honest position is more measured than the marketing around it.
Aspergillus species
Aspergillus is a very large genus, some members of which matter medically.
Serpula lacrymans
Dry rot is a wood-decay fungus rather than a surface mold, and it is the one entry here that is a structural rather than an air-quality problem.
Clean it, fix the damp
Common indoor molds. Clean the growth, fix the damp, and they do not return.
Cladosporium species
Cladosporium is the most common mold on earth and the most common one indoors.
Penicillium species
Penicillium is the mold on forgotten bread and the source of penicillin.
Alternaria species
Alternaria is one of the most significant airborne allergens there is and a well-documented asthma trigger.
Various surface fungi
Mildew is a loose term for superficial surface growth rather than a species, and it is what the overwhelming majority of bathroom 'black mold' actually is.
Various, often early-stage colonies
White mold is a description rather than a species, and much of it is the early stage of a colony that will turn green, grey or black within weeks.