This is a great question. You are referring the to Republican period of Roman history. Let me offer you some context.
At first Rome was a monarchy; in other words, they had kings. When the kings became corrupt, the people revolted and drove out the kings. When this happened the Romans needed a new form of government and this is when they started the Republic.
The Republic has magistrates who acted as checks and balances with respect to one another. In other words, there was a system of plurality and parity. For instance, there were two consul, who were the main leaders of the Republic. They did this, so that no one person would have absolute power. Besides the consuls, there were praetors and quaestors and other lesser magistrates. They, too, were based on the principles of parity and plurality.
In short, the Romans were phobic of the idea of kingship. In light of this, they had a Republic.