I cannot agree with the above answer. The old aristocratic families, the patricians (literally "elders") did exercise considerable power and influence; but they did not do so to the exclusion of other groups. The balance of the population were generally grouped into the Plebians. Although originally the plebians had no voice in public affairs, the class tension that arose from their exclusion led to the adoption of a constitution that gave them a voice in government. They were given the right to elect tribunes who represented their interests. Originally, there were two tribunes; but over time this number was increased to ten. Tribunes could veto any action of the Senate if they deemed it unfair.
Other provisions in the governmental structure protected the rights of the Plebians. Consuls were Patricians who had to be at least 40 years of age and have had military and governmental experience. they had full authority over foreign affairs; however there was a special legislative body known as the Concilium Plebis which was comprised entirely of Plebians and had full authority over domestic affairs.
By giving all classes a voice in government, the Roman Republic largely managed to avoid class conflict; however during those times in which there was a military or civil crisis, a dictator could be appointed who had absolute power for six months.
Particularly in the early days of the republic, the "democracy" was run almost entirely by aristocratic families that could trace their family roots back to the founders of Rome and derived their power from that lineage. There were laws that were outlined in a very comprehensive constitution that included separation of powers into distinct branches.
The rule of law, however, did not mean that ordinary Romans were able to influence the halls of power in any significant way. Despite the fact that there were elections and other avenues that suggested rule similar to our ideal of "by the people, for the people, and of the people," the reality was that the powerful families controlled the formation of and enforcement of the laws.
It was also true that even after many of the aristocratic families lost power that the common folks did not gain power rather it just shifted to a different type of aristocracy.