The curia per antonomasia was the Curia Hostilia in Rome, which was the building where the Senate usually met. The Senate, initially just a meeting of the city elders from all tribes (its name comes from "senex", which means "old man"), saw its powers grow together with the conquest that brought a town of humble origins to rule a large Republic (and then decrease steadily with the advent of the Empire).
During their expansion, the Romans exported the model to every city that gained the status of Municipium, so that it had its own Senate and its own officials charged with local administration (although they weren't usually elected but nominated by the central government; the only place where officials were actually elected by the people was Rome itself, and by Imperial times even those elections, although kept for the sake of tradition, had no more significance)(and Senators were never elected anyway, but had the position by family inheritance, like hereditary peers).
By the Imperial period, a curia was any building where local government held office, i.e. judicial proceedings, government meetings, bureaucracy, etc., and shortly afterwards the term started to refer also to the people making up the local administration.
Patriarchates and Major Archiepiscopates of the Eastern Rite have an assembly called the Patriarchal Curia, which assists the Patriarch or major archbishop in administering the sui juris church.
The Holy See retains an assembly called the Roman Curia, which assists the Pope in governing the Latin patriarchate and the entire Roman Catholic Church.
All of these have now 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 where subsumed by the only solid institution left, which was the church. The Bishop and its clergy basically took the place of the officials that the government used to send, to the point of actually sitting at the same chair in the same building. So the Curia passed in 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.