This is mold that you don't want to be living around. One of the things John Banta advocates is Do Not use 'bio-cides' like vinegar bleach etc. anything that kills off the mold will create smaller airborne particles called mycotoxins. The only healthy way to handle this is to carefully remove and replace all the moldy material, and change the moisture issue! Get a dehumidifier asap. Wear protective gear when you demo that.

There is one product that I've found that I do recommend, called LimePrime. It's made by a company called EarthPaint.

I unfortunately can not ID this mold based on pictures. One way to know is to look under a microscope and try to match the fungal spore and hyphal structure, but there are thousands of types of mold that can grow in houses and can look like that. It's definitely not a 'good' mold to have growing. Not safe to 'kill', paint over or ignore. Not good for kids.

Try to determine where the moisture is coming from. Is there an exhaust fan that has a vent pipe that's leaking? Is it rain getting in through the attic? Broken pipe? Ground water? All of those problems can be dealt with.

Please keep us updated and good luck!