Every year the people elected 2 Roman consuls
After the last king had been expelled, 2 Roman consuls were elected every year by the comitia centuriata. These men were equal in power, in such a way that the one consul could undo what the other had ordered.
This constitutional power was called imperium.
In the city the consuls would wear a purple-bordered toga (toga pretexta) and sat on a folding chair adorned with ivory (sella curulis).
The Roman calendar was based upon these consuls: The years were not numbered but indicated as the year of the consuls X and Y. For this the names of the consuls had to be registered in a sort of list (Fasti), the contents of which are available here.
The Roman consuls mainly acted as military commanders.
It was the senate that told them who had to go where with his army and provided the funds. Consul was the highest rank in the Roman society.
Life and death
In public the Roman consuls were preceded by 12 lictors, each of whom carried a fascis. Fasces were bundles of metal rods tied together with a red thong.
When outside Rome there was put an axe in the middle of the bundle. But see Dionysius of Halicarnassus The rods symbolized the power of punishment of flogging and the axe the power to the death penalty to be imposed by the Roman consuls.
Livius (8.7): Where consul T. Manlius has his own son beheaded for disobedience.
In the city
This implies that within the city the Roman consuls had no ultimate power over life and death. Indeed this power came ultimately to the people of Rome.
Livius: (2.29)(After the discontent plebs refused to be levied for the army, which was a crime, the same people would acquit the offenders, if the consul would condemn them)
As a rule a Roman consul could not be elected again within the next ten years. In times of crisis this rule was often ignored, because obviously an experienced commander in chief was needed. When the crisis was very deep one of the consuls could appoint a dictator. The senate could order the consuls to appoint a dictator, but the decision remained theirs. In the late republic the senate could appoint a dictator herself.
When the wars became more complicated and further from home, the Romans needed sometimes more than two consuls. In those cases they appointed one or more proconsuls. These men received the same imperium (power) as a consul but only in the area where they were supposed to operate.
In the imperial period the office of consul was not abolished, but the real power of course, was in the hands of the emperor.