The Roman Republic officially ended with the establishment of the Principate in 27 BC. Prior to that, legislative aspects of the Roman Republic were formed through the Comitia Centuriate (Century Assembly), the Comitia Tributa (Tribal Assembly), and the Concilium Plebis (Plebeian Council). These bodies operated on direct democracy, with "tribes" being formed like Congressional districts and into voting blocs. In the Centuriate Assembly, your assignment to a tribe was based on wealth, and it was skewed heavily towards the wealthier land owners. In the Comitia Tributa, it was solely on geography, and in the Plebeian Council, it was solely based upon geography and excluded patricians. These bodies had the sole authority to form legislation. The Senate functioned more as a judiciary function similar to the British House of Lords. It had some lawmaking ability, but was mostly constrained towards interpreting and striking down laws.

During the 2nd Century, the demographics largely shifted, and due to military and legal reforms, most of the population, particularly within the military, became disenfranchised and the democratic balances within the Republic failed. This lead to turbulence and ultimately Civil War.

Julius Caesar, who headed the Populare, or Populist Faction, won the ensuing civil war with Pompey Magnus, who represented the Optimate, or Conservative Faction. Julius Caesar then styled himself as a dictator, and ruled for ten years before his assassination. He was beloved by the commoners for his patronage, but reviled amongst the middle and upper classes for his authoritarianism and capriciousness. His assassination triggered another Civil War, between his assassins (Brutus and Cassius, and the rest of the Senate) and his successors (Octavian and Marcus Antonius).

With the victory of his successors, they continued autocratic rule, but Octavian attempted to consolidate his power by compromising significantly with the Senatorial Leadership in Rome, and including them within governance. Mark Antonty, by contrast, ruled a provincial government in the East, and aligned himself with Cleopatra, who headed the nominal protectorate state of Egypt.

After the final Civil War, Antony and Cleopatra were swept away and Octavian formally annexed Egypt, and used his vast revenues to keep the military loyal to him, rather than the Senate.

This would have precipitated a new civil war, but Augustus compromised heavily with the Senate, and in addition to being granted significant power (i.e. be a member of the Senate, with Absolute Veto, with full loyalty of the legions, above legal prosecution, and having the ability to directly appoint governors of 2/3s of the Provinces), he compromised by cleaving the Assemblies of their legislative authority and handed it over to the Senate.

This system proved very stable as it was essentially an elective monarchy that kept the window dressings of a Republic that lasted until the 230s AD, where the Roman Empire was nearly torn apart and it was replaced with an overt Monarchy.

The Roman Empire, however, collapsed because of Germanic migration and the collapse of the Western Empire in 476. The exact timing of this collapse is complicated, because the Eastern Roman Empire lasted until 1453, so Rome didn't "collapse" then. Additionally, its further complicated, because most Roman institutions remained intact, and the Germanic tribes, particularly the Ostrogoths in Italy, styled themselves as Viceroys, Regents, or Magister Militium (military Commanders) of the Eastern Roman Empire. This didn't sit well with the Eastern Capital (Constantinople), but they were unable to change the situation in their favor.

From 476 to 630, there were strong ties between the Western Germanic States (Visigothic Spain, Ostrogothic Italy, and Frankish Gaul), and the Eastern Roman Empire. There were strong cultural links that can be demonstrated through trade routes and accounts. Both had a lingua franca (Latin), common religion (Christianity, with some interspersing of Arianism, which still had residual support in the East), common coinage (the Solidius and Denarius), institutional similarity (Germanic Kings styled themselves after the Eastern Roman Empire and maintained similar bureaucracies).

This changed in 630, after the devastating Byzantine-Sassanid Wars, and also after the Islamic Conquests. The latter more so. The Islamic Conquests cleaved off Syria, Egypt, North Africa, and Spain from the Eastern Roman Empire and ended their naval dominance. Italy also fell to Lombard and Frankish invaders, and European Civilization genuinely lost institutional coherence.

After this, there were dramatic reforms within the Eastern Roman Empire, relating to Iconoclasms (which lead to schisms with the Papacy), the military (the Comitatenses and Professional Military were replaced with the semi-feudal Thema System), and the European Christian cultures split firmly between a Latin and Greek core. Though Latin had remained a spoken Language in Constantinople until the 1200s, it was relegated purely to academic circles and had fell out of the vernacular by the mid 500s.

Are you confused about what the difference is between the Roman Empire and the Roman Republic?

Which one was more of a better place to live?