For the republican period and the first few hundred years of the Empire, Roman jurisprudence was split in the provinces. Provincial people were accountable to their own legal systems so long as they were loyal to Rome and paid their taxes on schedule. The most famous historical example of the overlapping legal systems of the Empire was the biblical trial of Jesus before the Roman governor Pontius Pilate. Pilate tried to hand the case off to the local Jewish puppet king, Herod, who in turn refused it and handed Jesus back over to Pilate. In the end, Jesus was executed by the Roman government for inciting rebellion, using the traditional Roman punishment of crucifixion.
Roman citizens could always appeal to Roman law if they wanted to, even if they were part of provinces far from Rome. This changed dramatically in 212 CE when the emperor Caracalla extended citizenship to all free men and women (to make it easier to collect taxes). This was an important event because it extended Roman law to almost everyone in the empire; citizens were also exempt from some of the crueler punishments including crucifixion.
Some of the concepts and practices of Roman law were to outlive the empire itself. Rome initiated the tradition of using precedent to shape legal decisions, as well as the idea that there is a spirit to laws that is sometimes more important than a literal interpretation. The Romans were the first to codify the idea that someone accused of a crime was innocent until proven guilty; this was a totally radical idea in the area of justice, which in the rest of the ancient world normally held the accused guilty unless guilt could be conclusively disproved.