A consul was one of the highest offices in Republican (by election) and in Imperial Rome (appointed). The office is said to have existed since the expulsion of Rome's last king. Originally Consuls had religious and military duties. The rich could become one after 40 years of age & the lesser classes after becoming 42. There were two for each year & serving a year. If a consul died another was elected / appointed : a Suffectus. My list does NOT carry Suffecti (I believe the Wiki-lists do).
With the coming of Augustus the office became more an honor than a need & and soon the ruling Imperator was a Consul at the same tim ("Look Ma & I can do all in one day"). More and more Consuls resigned after some time that year making room for Suffecti. By Commodus' time doing the consulship could take only a month. Ofcourse also age was less and less imporatant.
With Constantine I Magnus one Consul started 'doing' Rome and the other Constantinople. So that after the definitive division of the Roman Empire & only one was left for each half. By now consuls could stay on for many years & though it is not known if it was a 'real' office by this time.
After 534 here were no more Consuls of what was left of Rome. Thriving Constantinople forgot it's use after AD 541.