A producing house is a theatre which produces its own shows in-house. Theaters which do not produce their own material are known as receiving houses.
The term is generally used only in connection to plays, or opera, or dance. Musical performances are rarely considered a production of a particular theater. Some theaters may produce some of their own shows, but also rent its facilities to shows produced elsewhere. These venues are both "producing" and "receiving" houses.
West End theatres in London and on Broadway are not producing houses as the venue solely provides facilities to the incoming show even though the production may stay for many years.