It is important to understand how society was arranged before trying to understand the political bodies of Ancient Rome. Government was built up from the family unit to the king with a logical order... 

The Components of Government

The Family: The lowest level of government was the family unit and the head of the family was the father, who had the power of life and death over all members of the family. The family also could have clients who the father was a patron of and slaves who the father was the master of. The family was the corner stone of roman society.

The Gens: A number of families with common ancestors formed together into a gens, all members of a gens were bound together by common religious rites and family ties. Each gens was governed by a decurio or chief who performed the religious rites and led the people into war.

The Curia: A number of gentes formed together into a curia. It was customary that all the gens in the curia worshipped a common god (even if they wouldn't naturally worship that god). A curio was appointed to preside over the common worship and to act as the military commander. the curio was assisted by a council of elders and he would consult them in the comitium before making important decisions.

The Tribe: Sometimes a collection of Curiae would join together for the purposes of defence or some other common purpose this was known as a tribe. Each tribe would have a rex or king who was a priest of the common religion, military commander and civil magistrate. He also had a council of elders he could call on to aid him in making important decisions. Over time this would lead to the formation of a society, this is believed to be what would have happened on both the Palantine and Quirinal hills. 

The Bodies of Government

At the time of the roman kings there were three main government bodies, these were

The Senate: This was the council of elders for the city. It was composed of 100 men selected by the king from the chief men of the gentes. The senate had no legislative power at all, it was a body that existed with the sole purpose of giving advise to the king, What he did with this advise was his own choosing.

The Comitia Curiata: This was the assembly for the city. It was made up of 10 curiae who would represent their own curia. Each man had one vote and the decision was made on the side with the majority. It is uncertain what the power of the assembly was during the time of kings. It was responsible for electing the king and passing laws to give him power but there is speculation that it also ratified or rejected the kings proposals though this remains unknown.

The King: The king was an elected for life official of the Roman people, he was the leader in every way, he was the chief priest of the national religion, the military commander of the people and the administrator of law and justice. Also like the father of the household he had the power of life or death over all his subjects.

When the two tribes on the Quirinal and Palatine hills merged their governments also merged. Their kings were picked alternatively from each tribe and their senate doubled to contain 200 members and the Comitia Curiata similarly doubled to 20 members. When the third tribe on the Caelian hill merged with the other two tribes an additional 100 senate members and 10 curiae were also added.

The Plebian Class

Rome was originally a merging of three tribes, each with equal power in Government. Each of these tribes members could become members of the senate or assembly or serve in the army. These old families and their descendents were called patricians and they formed an aristocratic class, exclusive and self serving.

 The result of this was that any people who were not a descendent of one of the three original tribes had no say in government nor could they fight in the army they were also unable to marry into the patrician families (by law) though they could own land. These people were known as the plebians. By the time of the Etruscan kings these people had become quite numerous they consisted of conquered people from other cities bought to Rome or people who had emigrated to Rome. As they became more numerous and some of them wealthy they wanted to be equal to the patricians.

Tarquinius Priscus, the first Etruscan king, is believed to have made the first steps to introducing the plebians into the state. Initially he introduced the more wealthy plebian families into each of the tribes under the name of lesser gentes while the old patrician families became known as greater gentes, starting to break down the line of separation between patricians and plebians.  

This change only affected a very small number of plebians and the majority were unaffected. The king Servius Tullius realised that Rome would be a stronger city if the plebians were allowed to fight in the army and had to pay taxes (which only the patricians were allowed to do at that stage). He decided to make new divisions of the people based on domicile rather then birth. He split the roman territory into twenty districts and all people, whether patricians or plebians who had settled homes were enrolled in the appropriate tribes and became subject to military service and tax. The people were also split into 5 census classes based on their wealth, these divisions were used to assign military duty and the different type of armour the the classes had to provide for themselves. 

This rearrangement of the people was intended to increase the size of the army and revenue to the government. Soon it also became a political body as well as a new assembly was formed called the comitia centuriata (assembly of the centuries) which met on the field of Mars outside the city walls. The voting was by century (the votes of the people of a century were counted to find what the century voted for and then the votes of each century was totalled). The first class had more centuries then the lower classes and accordingly had more say in the final outcome.

More detail and also read the lecture notes about the doubts of whether this actually happened.   

Over time the comitia centuriata gained the power of a real political and legislative body, of greater importance then the old comitia curiata. Incorporate the below

Even down to the end of the Republic a vestigial form of the old curiate assembly (the older assembly based on hereditary descent and said to be founded by Romulus) had to confirm the real elections of magistrates with imperium carried out in the centuriate assembly, and this indicates that the voting in the centuriate assembly was originally a usurpation of the powers of the curiate assembly, which was compelled to ratify the actions of the army. The curiate assembly was based on the old curiae, which were in turn subdivisions of the old tribes based on descent. Presumably, then,the curiate assembly was the normal assembly of the early kingdom, and its power were then usurped by the centuriate assembly that was represented the men in the army and was based on its military units. Could this new centuriate organization be connected with the fact that Servius Tullius had had no lex curiata passed to confirm his own imperium? Was he nominated then by the army and forced upon the civilian curiate assembly by the army assembled in its interal organization consisting of centuries? Certainly, the curiate assembly was moribund by the time of the creation of the office of censor in 445 or 434 BC. The election of the censors was cofirmed by a vote of the centuriate assembly itself, which shows that the curiate assembly was already an archaic vestige that could no longer be adapted.