In Roman Catholicism, a curia consists of a group of officials who assist in the governance of a particular Church. These curias range from the relatively simple diocesan curia, to the larger patriarchal curias, to the Roman Curia, which is the central government of the Catholic Church.
Other bodies, such as religious institutes, may also have curias. For example, the Legion of Mary has a rank called the Curia. It stands above the Praesidium but below the Regia. The Curia is responsible for several Praesidia.
All of these curias now have very different functions from the Curia in Roman times, but they keep the name since they are historically descended from it. In other words, when the Roman Empire collapsed, many of the administrative functions previously done by the state were subsumed by the only solid institution left, which was the church. The Bishop and curia took the place of the government officials to the point of actually sitting at the same chair in the same building. So the Curia passed into religious hands, and afterwards changed functions many times but always keeping its traditional name, at least in those Christian denominations that still keep a strong continuity with the Apostolic tradition.