Commodus ruled for 12 years, a much longer period than alluded to in the film. The film is wrong on this count. A republic is a system of government which does not have a hereditary monarch. An emperor is a monarch. The United States is a republic, and England is not.
Legend has it that Rome was originally ruled by Etruscan kings.
The first king was Romulus. The kings were overthrown in a revolution, which was sparked by the rape of Lucretia, in 509 BCE, by Sextus Tarquin, the son of the seventh and last king, Tarquinius Superbus. Dictators and kings were thereafter despised by Romans, hence, the ideological adulation of a republican system of government, which was a central theme of both Roman history and the movie. After Commodus was murdered, the Senate met before daybreak, and declared sixty-six year old Pertinax, who was the son of a former slave, emperor. Pertinax thus became emperor on January 1st, but he was murdered by a group of soldiers the following March, after less than three months in power.