This is a great question. At first Rome was a monarchy. However, due to the corruption of the final few kings of Rome, they were expelled. At this point Rome stated a new form of government. The traditional date for the start of the Republic is 508 B.C.
The Republic was a form of government based on checks and balances. The two highest officials were called consul. They were elected officials and they held supreme power for one year. The Roman invested them with what they called "imperium," that is power to rule, govern, and lead an army.
The consul usually came from the highest social order and it was a mark of incredible prestige. Whenever someone became consul from a non-aristocratic lineage (that is, there were no past consul is his line), he was considered a new man (novus homo). For example, Cicero, the great orator, was one of them.
People could become consul more than once, but the competition was fierce, as you can imagine. Someone like Marius was consul seven times. In time, the election process at Rome became more and more corrupt and eventually the Republic was destroyed through civil wars and the Empire started.
During the Empire there were still consul, but they no longer had the power and prestige of former times. The Emperor eclipsed everything.