Part 1:

This answer is likely more detailed than maybe you were expecting but it evolved and expanded as I wrote it, which is at least thematic for this topic.

Let me answer your question by breaking down how the Roman Republic actually governed itself and the institutions that formed its makeup.

Polybius, the most prominent Greek historian for Roman history (especially in the 3rd and 2nd centuries BCE), believed that there were several reasons why Rome was able to grow and dominate so quickly. For our purposes we will focus on his commentary on the Roman state, specifically its balance and stability. To him, other city-states, kingdoms, and empires were so internally unstable that they could not hope to rival Rome’s ability to focus on external and foreign affairs. This, he believed, gave Rome the edge in warfare. Polybius’ commentary draws our attention to Rome’s Constitution, which, importantly, was not written down anywhere. Instead, it was a product of development based on a series of social and political struggles that led the whole citizen body (by this I mean the male enfranchised population) to make various compromises to maintain order. Since the constitution wasn’t written down, a system of agreed-upon practices of doing things formed. Collectively everyone knew it and supported it (very few questioned it). For instance, in the Struggle of the Orders, the plebs effectively broke away from the Roman state and shaped their own counter-state that forbade the inclusion of Patricians. It was made up of a Council of Plebs, Assembly of the Plebs (sometimes called the Tribal Assembly), and had its own officers in the form of Plebeian tribunes and aediles (maybe this answers your question but I think you are looking for something far later). When this Plebeian state was later amalgamated into the Roman Republic it gave it an entire section of its government that wasn’t there beforehand; notably, none of this was written into the constitution but instead it just happened.
This constantly changing and evolving state of affairs brings me to discussing the portrait of the Roman Republic’s government. Let me state here that this is going to be a conflation, a broad description of the system around the years c.200-150 BCE onwards. The shape of the Roman state is a mirrored reflection of themselves as a people: dynamic and adaptive, but also extremely conservative. The Romans probably called their state the mos maiorum (“the way of the ancestors”) as they tended to fall back on old traditions to address issues or conduct governance as their forebears and elders did. However, they would modify the system when circumstances demanded (as noted with the detaching and merging of the Plebeian state). This clash between tradition and innovation created many redundancies in their system of administration which we will explore soon.

In the broadest sense the Roman state was divided into two parts: the first being the meetings of citizens and the second being the officers attached to these meetings. Many of these officers joined or were attached to the Senate and were called magistrates. Now two sets of different officers, technically not magistrates, were attached to the Assembly of the Plebs. These were the Plebeian tribunes and aediles (the very same assembly and officers sprung from the Struggle of the Orders). Now all these officers were elected by popular vote in a popular assembly and usually held office for a year. The Senate (“the council of the old men [or men with experience]”) was the leading meeting in the Roman state due to tradition and circumstance. This mostly happened as a result of Rome’s many wars and foreign threats; the generals, the magistrates, the educated, and the wealthy all collected in this body and the population began to depend on it more and more for leadership, experience, and expertise especially in the wake of the First and Second Punic Wars. Thus, through consensus of the people it became preeminent as Roman citizens began to depend and refer to it for guidance on political matters. However, it should be stated, it had no legal power. It could not pass laws. Instead, its function was advisory, to issue their senatus consulta (“pieces of advice by the Senate”) on issues it had debated as guidance for the magistrates; these were not legally binding. So to summarize, the Senate made decrees (pieces of advice to magistrates), which were given to officers who could proceed to take these decrees to the popular assemblies to be voted on and made into law. However, due to the dominance of the Senate in the state, its decrees were customarily taken to the assemblies and passed without an issues since officers rarely wanted to go against the wishes of fellow senators and the people usually trusted in the wisdom and experience of their elders and “betters”.