Imperium in a broad sense translates as power. In ancient Rome the concept applied to people, and meant something like "power status" or "authority", or could be used with a geographical connotation and meant something like "territory". It is not to be mistaken with auctoritas ("authority").
Imperium can be distinguished from regnum, or royal power, which was inherited. Imperium was originally a military concept, the power of the imperator (general in the army) to command. The word derives from the Latin verb, imperare ("to command"). The title imperator was applied to the emperor, who was the commander of the armed forces. In fact, the Latin word, imperator, gives us the English word emperor.
Imperium was indicated in two prominent ways. A curule magistrate or promagistrate carried an ivory baton surmounted by an eagle as his personal symbol of office (compare the field marshal's baton). Any such magistrate was also escorted by lictors bearing the fasces (traditional symbols of imperium and authority); when outside the pomerium, axes were added to the fasces to indicate an imperial magistrate's power to enact capital punishment outside of Rome (the axes were removed within the pomerium). The number of lictors in attendance upon a magistrate was an overt indication of the degree of imperium. When in the field, a curule magistrate possessing an imperium greater or equal to praetorian imperium wore a sash ritually knotted on the front of his cuirass. Further, any man executing imperium within his sphere of influence was entitled to the curule chair.
As can be seen, dictatorial imperium was superior to consular, consular to praetorian, and praetorian to aedilician; there is some historical dispute as to whether or not praetorian imperium was superior to "equine-magisterial" imperium. A promagistrate, or a man executing a curule office without actually holding that office, also owned imperium in the same degree as the actual incumbents (i.e., proconsular imperium being more or less equal to consular imperium, propraetorian imperium to praetorian) and was attended by an equal number of lictors.
Certain extraordinary commissions, such as Pompey's famous command against the pirates, were invested with imperium maius, meaning they outranked all other owners of imperium (in Pompey's case, even the consuls) within their sphere of command (his being "ultimate on the seas, and within 50 miles inland"). Imperium maius later became a hallmark of the Roman emperor.
Another technical use of the term in Roman law was for the power to extend the law, beyond its mere interpretation, extending imperium from formal legislators under the ever-republican constitution: popular assemblies, senate, magistrates, emperor and their delegates to the jurisprudence of jurisconsults.
Thus absolute, universal power was vested under early Islam in the original Caliphate (before it became the political toy of worldly powers 'behind the throne' and was even politically discarded by essentially secular princes), and later again claimed by Mahdis.
While the Byzantine Emperors retained full Roman imperium and made the episcopate subservient, in the feudal West a long rivalry would oppose the claims to supremacy within post-Roman Christianity between sacerdotium (the 'priesthood', i.e. the clergy ministering the word and will of God) in the person of the Pope and the secular imperium of the revived western Emperor since Charlemagne. Both would refer to the heritage of Roman law by their titular link with the very city Rome: the Pope, Bishop of Rome, versus the Holy Roman Emperor (even though his seat of power was north of the Alps).
The Donatio Constantini, by which the Papacy had allegedly been granted the territorial Patrimonium Petri in Central Italy, became a weapon against the Emperor. The first pope who used it in an official act and relied upon, Leo IX, cites the "Donatio" in a letter of 1054 to Michael Cærularius, Patriarch of Constantinople, to show that the Holy See possessed both an earthly and a heavenly imperium, the royal priesthood. Thenceforth the "Donatio" acquires more importance and is more frequently used as evidence in the ecclesiastical and political conflicts between the papacy and the secular power: Anselm of Lucca and Cardinal Deusdedit inserted it in their collections of canons; Gratian excluded it from his Decretum, but it was soon added to it as Palea; the ecclesiastical writers in defence of the papacy during the conflicts of the early part of the twelfth century quoted it as authoritative.
In one bitter episode, pope Gregory IX who had several times mediated between the Lombards and the Hohenstaufen emperor Frederick II reasserted his right to arbitrate between the contending parties. In the numerous manifestos of the pope and the emperor the antagonism of Church and State becomes daily more evident: the pope claimed for himself the imperium animarum 'command of the souls' (i.e. voicing Gods will to the faithful) and the principatus rerum et corporum in universo mundo 'princedom over all things and bodies in the whole world', while the emperor wished to restore the imperium mundi, imperium (as under Roman Law) over the (now Christian) world — Rome was again the be the capital of the world and Frederick was to become the real emperor of the Romans, so he energetically protested against the world-empire of the pope. The emperor's successes, especially his victory over the Lombards at the battle of Cortenuova (1237), only embittered the opposition between Church and State. The pope again excommunicated the "self-confessed heretic", the "blasphemous beast of the Apocalypse" (20 March, 1239) who now attempted to conquer the rest of Italy, i.e. the papal states, etcetera.
The chief minister of Henry VIII, the Archbishop of Canterbury Thomas Cranmer suggested removal of the Roman Catholic papacy's imperium in imperio (Latin equivalent of state in the state) by requesting that Parliament pass the Act in Restraint of Appeals (1533) specifying that England was an empire and that The Crown was imperial, and a year later the Act of Supremacy proclaiming the Imperial Crown Protector and Supreme Head of the Church of England.
In Orthodox Russia too, when Peter I the Great has assumed the Byzantine imperial titles Imperator and Autokrator, instead of the 'merely' royal Tsar, the idea in founding the Russian Holy Synod was to put an end to the old Imperium in imperio of the free Church, by substituting the synod for the all too independent Patriarch of Moscow, who had become almost a rival of the Tsars — Peter meant to unite all authority in himself, over Church as well as State: through his Ober-Procuror and synod, the Emperor rules his Church as absolutely as his army and navy through their respective ministries; he appoints its members (mostly bishops) just as his generals; and the Russian Governments continued his policy since.
Even in 19th century North America, when by the decree of the President of the United States, Brigham Young, the Mormon hierarch and head of the Church of the Latter Day Saints, was appointed first Governor of the Territory of Utah on 28 September, 1851, this was called (politically, not in law) establishing a theocratic form of government there (until it became a regular state) as an imperium in imperio, within the limits of the republic.