Who can adjust a person's immigration status in the USA?

This question is not very clear, but generally people who are in the U.S. with a valid nonimmigrant visa and then become eligible to adjust status in one of the many immigrant visa categories, may go through the adjustment of status process in the U.S. This generally requires that a visa is "immediately available" for the person at the time the case is filed. Determining whether a visa is "immediately available" can be discerned by review of the current Visa Bulletin issued by the U.S. Department of State. Some visa categories (such as EB2/NIW and EB1 Extraordinary Ability) are "self-sponsoring," others require a qualifying family relationship or sponsorship by a qualified employer. If you are asking about eligibility based on some specific family relationship or some specific job, it would be wise to consult with an immigration attorney who could learn all of the relevant facts and then advise you about eligibilities, options and strategies. Some immigration law firms, including mine, offer legal services on a "flat fee" basis so that a client will know the total expense from the very beginning, and a few immigration law firms, including mine, offer an initial consultation free of charge.

USCIS does it; if the person in removal, then a judge.

There are many circumstances when this can be done. It is quite complicated and would definitely require an initial consultation with our office.