Short answers, pulled from the story.
The patron-client relationship in ancient Rome was called clientela. The patron was called a patronus and the client a cliens. The legal term for the protection a patron offered was patrocinium.
The salutatio was the daily morning reception at which Roman clients visited their patron's home to make requests. The patron received clients at dawn in the atrium and tablinum, after which the clients escorted him to the forum.
When a slave was manumitted, the former owner became their patron and the freed person, called a libertus, owed obligations including campaigning on the patron's behalf, carrying out jobs or errands, and in some cases continuing a sexual relationship that had begun in servitude.
Traditional clientela began to lose its importance as a social institution during the 2nd century BC. The historian Fergus Millar has argued that clientela was probably not the dominant force in Roman elections that it has often been considered.
Augustus established colonies throughout the empire during his conquests and distributed food and money to the people of Rome. He also settled soldiers in new colonies he personally sponsored, placing large numbers of people in his debt and extending clientela to an imperial scale.
Caesar restored the Aedui's influence over other Gallic tribes who had once been their clients. After that, the Aedui called on him repeatedly to perform the duties of a patron, and many in Rome came to regard him as the patron of the Aedui.