This question is pretty big. Keep in mind that ancient Rome is a 1200+ year time span. So I'll be super-general.

The big thing is that Rome always treated citizens and non-citizens differently. Only non-citizens could be crucified. Hence, the story goes, that when Peter and Paul were brought to trial in Rome, Peter was crucified and Paul was beheaded because he was a citizen. Roman citizenship was given to the children of citizens and freed slaves of citizens and to communities outside of Rome as an honor. It spread to all of Italy in 80 BCE and kept expanding until everyone in the empire was granted citizenship in the second century CE.

Citizens could be executed but only after a public vote. That tradition dated all the way back to the monarchy. Verres was basically run out of Rome for supposedly executing a Roman in Sicily and Cicero was likewise exiled for hanging the members of the Catilinarian conspiracy after a vote of the Senate without a public vote.

As was the case for Cicero, the public vote was often done away with when the Senate believed that the threat was too large to wait. This provided a precedent for the proscriptions of Sulla, the Second Triumvirate and the empire, where people accused of treason were tried by a special court and could be killed.

Legal processes in the city of Rome were different for legal processes outside of the city. Within the city of Rome they used proceeding dating to the monarchy. Likewise each city of the empire had its own laws often of equal antiquity. However the Roman magistrates overseeing provinces would often arbitrate, in which case they had essentially unlimited authority. The only thing stopping an abuse of power was the threat of being brought to trial back in the city of Rome (which happened often but mostly to little effect).