Who can I not rent to if the apartment is attached to my primary dwelling

What do you mean by attached? Do you actually share space OR is the apartment separate?

Is this like a duplex or what?

Rental discrimination laws are federal. Therefore you cannot discriminate against any protected class.

What is prompting the question.

You do not have to rent to anyone that can not meet your payment terms ( they have to be able to pay you) If you do back ground checks you have to use same standard for all applications.

If it is a one bedroom, you can refuse to rent to any family larger than 1 or 2 people. You can refuse to anyone that does not follow any local zoning or code for number of people in a certain number of bedrooms.

Also you have to be sure it is a legal apartment and you are zoned to rent.