What could it be that has infested my brothers living quarters downstairs in my home. There are microscopic sort of biting things that are on all of his furniture and in the carpet, and in all of his clothes. We have had exterminators here and they have bombed his apartment, but the things are still there. He has to vacuum every day, and wash his clothes daily, still they are there and they also bite. He has been to the dermatologist several times, but they are no help. This has been going on for five months and it is driving him insane. These things however are not upstairs in my living quarters, what could they be and how do we get rid of them. We are desperate now and need help.

I do not want to say what is it, since without seeing them and knowing, it is hard to say. But some things like Mites may require couch and other items to be thrown out, the house sealed and fumigated.
Not just bombed.

Your brother needs to collect some samples and have them identified, as they could be any of several different species. An easy method of collecting them is to catch them on transparent tape then tape them to a white unmarked paper or card. A teaching Hospital or a University Biology Dept. Entomology lab are good places to get an I.D.

If the exterminator didn't bother to identify what he was treating for, let me recommend finding another exterminator.

Once you know what you have, you're half the way to getting rid of it.

IF you can't do as suggested, and catch some on tape and have them identified.

One tiny mite that is extremely difficult to get rid of is a bird mite.
Has he ever had a bird, or brought home items, clothes, or furniture from someone who might have?
Some people literally have to get rid of everything they own. Throw out what isn't absolutely necessary and put the rest in cold storage for several months, and have it fumigated under a tarp.

Edited to prioritize suggestions: Catsmine