Privately run means that it is owned and managed by a private company as opposed to being a government run institution. A detention center is where someone is held for a period of time - somewhat like a prison.

A "privately run detention center" is a detention center that is run by a commercial, private, company rather than by the government or state. i.e. it is a detention center that is privately run, where 'privately run' means 'operated by a private company' and 'private company' is the same as a commercial company that is not owned by the state/government.