Eparchy is an anglicized Greek word, authentically latinized as eparchia and loosely translating as 'rule over something', but has the following specific meanings, both in political history and in the hierarchy of eastern churches.

Secular jurisdictions

Tetrarchy model
In the linguistically often illogical, mixed Greco-Latin jargon of Byzantine administration, eparchy (or its re-latinization eparchia!) is mainly used as the 'literal' Greek version of the Latin praefectura 'prefecture', i.e. the office, term or resort (rather Latin provincia in the widest sense, not necessarily territorial) of any Praefectus,

Byzantium
The Roman title of Eparch, as governor of a province of Roman Greece, was also used as equivalent to, or represented that of the Roman praefectus. The area of his administration -prefecture- was called an eparchy. The term was revived as one of the administrative sub-provincial units of post-Ottoman independent Greece, the country being divided into nomarchies, subdivided into eparchies, again sub-divided into demarchies.

Russia and USSR
The Christian Church (before the split into Roman Catholic and Byzantine Orthodox) adopted the temporal administrative division since the Tetrarchy in the Dominate, and part of its terminology, as convenient for internal use, but adapted it as follows.
Under these patriarchates and exarchates came the eparchies under metropolitans; these had authority over the bishops of various cities. The original ecclesiastical eparchies then were provinces, each under a metropolitan. The First Council of Nicaea in 325 accepts this arrangement and orders that: "the authority [of appointing bishops] shall belong to the metropolitan in each eparchy" (can. iv), i.e. in each such civil eparchy there shall be a metropolitan bishop who has authority over the others. This is the origin of ecclesiastical provinces.
Later in Eastern Christendom, after a process of title-inflation, multiplying the numbers of dioceses, metropolities and (arch)bishops and reducing their territorial size, the use of the word was gradually modified and now it means generally the diocese of a simple bishop.
Thus in Eastern Orthodoxy, Oriental Orthodoxy, and Eastern-rite Catholicism, an eparchy is the jurisdiction of a bishop, corresponding to what in the West is called a diocese.
The name Eparchy was, however, not commonly used except in Russia, as the usual term for a diocese. The Russian Church in the early 20th century counts eighty-six eparchies, of which three (Kiev, Moscow, and St. Petersburg) are ruled by bishops who always bear the title "Metropolitan", and fourteen others under archbishops.

The Praetorian prefectures of Gaul, Italy (i.e. the whole western empire, largely in economic decline), and Illyricum made up the Roman Patriarchate, under the Pope
the Prefecture of the East was divided (in the fourth century) between the Patriarchs of Alexandria and Antioch and three deputies styled Exarchs.

• the Diocese of Egypt was the Patriarchate of Alexandria
  the Diocese of the East (not to be confused with the Prefecture of the East) became that of Antioch.
  the Diocese of Asia was under the Exarch of Ephesus
  the Diocese of Pontus under Cappadocia, and Thrace under Heraclea.