In very general terms, the government of the Roman Republic was made up of these three parts:
The Senate - Members in the Senate were often members of the richest and most powerful families in Rome. Terms were lifetime and they were chosen by the Consuls.
The Consuls - Two consuls were elected by the Assembly and acted as co-rulers.
The Assembly - This is where everyone else fit in, the "common" folk of Rome. Though they did not wield as much power as the Senate, if a Senator wanted to be Consul, he had to convince the Assembly because only they had the power to elect them.