Short answers, pulled from the story.
The Principate was the early period of the Roman Empire, beginning with Augustus's reign as the first Roman emperor in 27 BC. It conventionally lasted until the late third century, often linked to the Crisis of the Third Century, after which the Dominate began under Diocletian.
Princeps means 'leading citizen' or 'first citizen' in Latin. The title had previously been applied to Roman nobles who distinguished themselves in service to the Republic; Augustus refashioned it as the title for the sole ruler of Rome, while preserving its republican connotation.
Augustus accumulated power incrementally through a series of settlements with the Senate. On the 16th of January 27 BC, the Senate granted him oversight of Rome's most turbulent provinces and their legions. By 23 BC he had also received imperium proconsulare maius, authority superior to all other proconsuls across the whole empire, and tribunicia potestas for life. On the 6th of March 12 BC he became pontifex maximus.
Theodor Mommsen was a 19th-century German scholar who introduced the constitutional concepts of 'Principate' and 'Dominate' as a framework for Roman imperial history. He viewed the Principate as a diarchy in which power was formally shared between Augustus and the Senate from 27 BC, and considered it an extension of the Republic until its rupture in the 3rd century. Later historians, following Ronald Syme, rejected this view and called the Principate a variant of monarchy.
Roman law contained no explicit outline for how emperors should be chosen or who could become princeps. Augustus based succession on auctoritas, personal authority earned by merit. After the fall of the Julio-Claudian dynasty in AD 68, Vespasian established a firmer legal basis via the lex quae dicitur de imperio Vespasiani. Under the Antonine dynasty, emperors typically appointed a capable successor, but modern scholarship rejects the idea this was a deliberate policy. Commodus, whose 12-year reign began after Marcus Aurelius, was the first emperor's son born during a father's reign to inherit the position.
The Principate conventionally ended with the Crisis of the Third Century in the late third century AD. It was replaced by the Dominate, a more overtly autocratic system associated with the reign of Diocletian, which Mommsen had identified as the rupture breaking the Principate's republican forms.