In some parts of the world, pretty much every city had its own calendar at that time. A few calendar era systems had a wider spread, such as the Han calendar in China, and the Seleucid era calendar in much of the Near East. In the Mediterranean world the most important calendar era systems were those of the Roman consuls, the Alexandrian calendar, Eratosthenes' Olympiad count, and the Seleucid year count.

Many ancient calendars were what are called "eponymous" systems. In an eponymous system you refer to a specific year by citing the regnal year of the local ruler; or in places where magistracies were held for only one year at a time, by naming the holder(s) of the magistracy in that year (e.g. "during the consulship of Cossus Cornelius Lentulus Gaetulicus and L. Calpurnius Piso" in Rome, "during the archonship of Polycharmos" in Athens). The Chinese era system varied this idea at the time by having each emperor declare a new era every five years or so (so 1 BCE comes out as either 2 Yuánshòu or 1 Yuánshǐ, as there was a change of emperor that year).

The Olympiad, Alexandrian, and Seleucid systems are simpler to modern eyes. They adopted a simple numerical count from a specific year: so 1 BCE comes out as the 4th year of the 194th Olympiad, 748 Nabonassar/324 Philippos in Alexandria, or year 311 Seleucid.

The Roman and Alexandrian systems had additional quirks in that they had both recently undergone a conversion from a lunar calendar to a solar calendar (in 46 BCE for the Romans, ca. 30 BCE in Alexandria); this didn't affect the way you refer to years, but it did affect where those years began and ended. Most ancient calendars had the year beginning in northern hemisphere midsummer, contrary to the practice of the Julian/Gregorian calendar, so a lot of the time you'll see a year quoted as two dates: so e.g. we would write "the archonship of Polycharmos" as "2/1 BCE".

There's a good reason we don't usually bother quoting dates in ancient calendars.

Edit. moved an irrelevant example to a place where it's more relevant.

If you were Roman it would have been 753 Ab urba condita or 753 years since the founding of Rome. But only their historians used that.

They regularly referred to the years by who was consul. So whoever the two consuls were for 1bc and 1ad were how the years would be referred.

This question appears to be answered in several parts of the FAQ, specifically the first two sections of Calendars and Timekeeping

For example, How did civilizations determine what "year" it was before an "international" year was established? gives several examples, briefly, and the "Other dating systems" subsection has more.