Short answers, pulled from the story.
Imperium was a formal legal authority held by certain Roman magistrates and commanders to govern a military unit, province, or territory. It was distinct from auctoritas and potestas, which were generally considered inferior types of power. A man holding imperium had, in principle, absolute authority to apply the law within the scope of his office.
Imperium was signified by the number of lictors, attendants carrying the fasces, who accompanied a magistrate. A curule aedile had 2 lictors, a praetor 6, a consul 12 each, and a dictator 24 outside the pomerium. Magistrates also carried an ivory baton surmounted by an eagle and sat on the curule chair.
Imperium maius, or greater imperium, was an elevated form of authority that outranked all other holders of imperium within a given sphere. Pompey's command against the pirates was invested with imperium maius, described as ultimate on the seas and within 50 miles inland. It later became a defining feature of the Roman emperor's authority.
Curule magistrates and promagistrates held imperium. These included the curule aedile, the praetor, the consul, the magister equitum, and the dictator. Promagistrates exercising a curule office without formally holding it also possessed imperium of equal degree to the actual officeholders.
Both the pope and the Holy Roman Emperor claimed supremacy using the language of Roman imperium. Pope Gregory IX asserted the imperium animarum, the command of souls, while Emperor Frederick II sought to restore the imperium mundi, Roman authority over the Christian world. Gregory excommunicated Frederick on the 20th of March 1239 during their bitter struggle.
The English word emperor derives from the Latin imperator, which originally meant army commander in ancient Rome. The title imperator later became the exclusive designation of the Roman emperor as commander of the armed forces.