Short answers, pulled from the story.
A Roman censor was responsible for three main areas: conducting the census of all citizens and their property, maintaining public morality through the regimen morum, and overseeing state finances and public works. The census included enrolling senators, reviewing the equestrian order, and assigning citizens to tribes and classes.
The censorship was originally set for a full lustrum of five years, but a law proposed by the dictator Mamercus Aemilius Mamercinus limited the term to eighteen months. The censorship as a regular office existed for 421 years, from 443 BC to 22 BC.
Gaius Marcius Rutilus became the first plebeian censor in 351 BC. Twelve years later, in 339 BC, one of the Publilian laws made it mandatory that one of the two censors always be a plebeian. Both censors were plebeians for the first time in 131 BC.
The nota censoria was a formal mark of censorial reproach placed next to a citizen's name in the public lists, stating the cause of punishment in writing. It produced ignominia, a temporary reduction in status, but did not strip a man of his current magistracy or bar him from future office. Both censors had to agree before the mark was valid, and a subsequent pair of censors or a law could erase it.
The censorship declined after Publius Clodius Pulcher restricted its powers in 58 BC and civil wars left the office unfilled for years. Augustus appointed the last two censors, Lucius Munatius Plancus and Aemilius Lepidus Paullus, in 23 BC. After that, emperors performed the censors' functions under the title Praefectura Morum.
The Roman censor's power to brand citizens with a nota censoria for conduct judged harmful to public morality directly produced the modern meaning of the words censor and censorship. The authority to judge what was morally acceptable in public life transferred into the concept of officials who review and suppress material deemed contrary to public morality.