It's not exactly black and white. I don't know about the law in your state but where I live there are a number of crimes that would prevent a person to adopt, mainly violent and sex crimes but there are others as well.
However, not having a criminal background or just a minor criminal background does not mean that you will be approved to adopt. The agency handling your application will evaluate your fitness as a parent and will consider a number of things. You don't have to be perfect but you do have to meet a certain standard. If you want to adopt internationally the standards are pretty high and it's quite expensive. But if you adopt a waiting child domestically (a child in foster care) the standards are lower and it's significantly cheaper. The biggest cost in such an adoption is attorneys fees which the adoptive parent has to pay.
As a part of the adoption process the agency you are working with will do a background check, check your financial status, your family, your home, your kids and your character in general, etc. Other people in your household will also have to have criminal background checks done. They will inspect your home and interview you. The social worker that does your home study has some discretion in determining if you would be considered a good candidate for adopting a child.
