To start off, I suggest that we bear in mind an initial structure, based on a class system of King : Aristocracy : Plebs : Slaves. To this we have a strata of “clients” which we can more or less ignore for the sake of simplicity.
It is quite probable that the introduction of a republican system took some transition so that the first leaders, like Lucius Brutus who ousted the last king Tarquin the Proud, werede facto like kings although clearly they had to watch their backs from being accused of it.
With the elimination of the King, the obvious thing was for the aristocracy, the Patricians, to use their wealth and power to fill the vacuum. They did, and through control of the Senate (the council of elders/aristocracy) they made laws which tended to defend their power. As one-sided as this was the Plebeians eventually managed to shift the system towards something a little more egalitarian.
A number of institutions developed in order to give Patricians, Plebeians and the Military a voice in government. The government itself developed a bureaucracy and positions of varying power which were open to this or that social class.
Persons campaigning for political position would spend personal fortunes in order to win votes. Julius Caesar himself drove himself to near ruin when as Aedile of the city he organised great circus displays for the joy of the people. By the time he was moving on from that position to govern Spain he had many creditors on his heels and was only saved by his close friend Crassus – the richest man in Rome.
The Senate existed throughout all the periods of ancient Rome: Kingdom-Republic-Empire, although the power it wielded clearly shifted and at times was purely nominal rather than actual. At first its members were the Patrician heads of the different gens (clans) – the aristocracy of Rome. In later periods ex Consuls were included and finally even Plebeians.
In modern terms we might regard it rather like a House of Lords & Commons in the UK – a chamber where the members discussed, voted and emitted a common verdict about this or that issue. The Senate house was divided into two halves with the seats on one side or the other reserved for those for or against the particular issue being discussed.
During the times of the Republic the Senate had 300 members. During the times of the Empire the number grew to 600. Throughout this long period the real powers of the Senate were in fact decreasing so that by the end of the empire it was little more than a local council.
The Comitia Curiata
The comitia curiata was an assembly through which the citizens (initially only Patricians!) of Rome could participate in the government. It was in fact an assembly of the curiae (groups of patrician family-clans). As has already been mentioned, during the early days of the kingdom and republic there were 30 curiae in total and each had a single vote. Through the Comitia Curiata the citizens could propose laws, but these would have to be approved by the Senate in order to become valid. During the kingdom the assembly was in charge of ratifying the king’s nomination and if it wished it could even reject the nominee.
During the time of the Republic when the position of King was replaced by Magistrates and Consuls the assembly continued to participate in their election and ratifying of their position.
The Assembly of Centuries – a military thing
We should remember that the citizens of Rome belonged to the Patrician families. These families were grouped into gens (clans), the gens were grouped into 30 curia and the curia into three tribes. Each tribe saw to providing a set number of cavalry and foot soldiers, subdivided into groups of a hundred soldiers led by a Centurion.
The Assembly of Centuries evolved from the assembly of the tribes and was essentially military in nature although it had civil voting rights and powers. Given that the army was essentially made up of wealthy citizens only a new assembly was created – The Assembly of the People – which looked after the interests of the poorer part of society, mostly Plebeians.
The Assembly of the People (assembly of tribes/meeting of plebeians)
This was an assembly created in order to give the (landowning) plebeians a means of voting their own “anti-senators”, called Tribunes. They had the power of standing up for the rights of the Plebeians when confronted by the laws written by the Senate and Patricians. They guarded the rights of the individual against the rules of the state.