Short answers, pulled from the story.
The German scholar Theodor Mommsen proposed the term Principate in the 19th century to describe a constitutional system where imperial authority sat inside Republican institutions. Historian Jochen Bleicken credits Mommsen with introducing these concepts as a framework for periodizing Roman imperial history.
On the 13th of January 27 BC Octavian made a show of returning full power to the Senate and relinquishing his control of the Roman provinces and their armies. He accepted a ten-year responsibility of overseeing provinces considered chaotic on the 16th of January 27 BC.
During the second settlement Augustus was granted the power of tribune for life though not the official title of tribune. This power allowed him to convene the Senate at will veto actions preside over elections and speak first at any meeting while also receiving imperium proconsulare maius giving constitutional power superior to all other proconsuls.
With fall of Julio-Claudian dynasty in AD 68 principate became more formalized under Emperor Vespasian from AD 69 onwards. The position of princeps became distinct entity within broader formally still republican Roman constitution allowing civilian aspect of Augustan ideal to give way to military role.
This period saw several firsts including recognized period of rule by two emperors during co-rule of Marcus Aurelius and Lucius Verus from their accession in 161 AD until latter death in 169 AD. Succession of Commodus marked first transfer of power to son born to sitting emperor despite often ranking amongst worst Roman emperors.