The years to the Romans were distinguished by which consuls ruled the Roman Republic that year. For instance, an event that happened in 63 BC would be chronicled by a Roman historian as occurring in "The year of Cicero and Hybrida". A famous example mocking the practice is mentioned by Suetonius in The Twelve Caesars, when Julius Caesar ruled as consul with Bibulus in 59 BC, the latter stayed in his house throughout the consulship out of protest for the ill-treatment Caesar gave to Bibulus, Romans mockingly referred to events of his consulship as taking place "In the year of Julius and Caesar".

The "Anno Domini" system (that is, basing year 1 on the year of Jesus' birth) was popularized by the Anglo-Saxon historian Bede. Though I do not believe this was in common use, it appears that sometimes, before Anno Domini was incorporated by the Romans, they based year 1 on the legendary founding of Rome by Romulus in 753 BC.

Sources

“Notes and Queries: How They Counted the Years in the BC Era; What Noise Does a Giraffe Make?” The Guardian, Guardian News and Media, 13 Jan. 2010, https://www.theguardian.com/theguardian/2010/jan/13/how-ancient-rome-counted-years.

The Editors of Encyclopaedia Britannica. “St. Bede the Venerable.” Encyclopædia Britannica, Encyclopædia Britannica, Inc., 21 May 2019, https://www.britannica.com/biography/Saint-Bede-the-Venerable.

Tranquillus, Gaius Suetonius, et al. The Twelve Caesars. Penguin Books, 2007.