Individual cultures tend to develop their own calendars and means of measuring time. For Rome, dates were calculated "From the founding of the City," Latin Ab urbe condita: the city of Rome was believed to have be founded in what we consider 753 BC, meaning that that year for the Romans was 1 AUC. Likewise, Julius Caesar would have seen himself as being born in the year 653 AUC, rather than in 100 BC. Other societies, like Athens, Egypt, or Persia, would have developed their own calendars that they would have understood from major events in their own history, so births of important figures, the creation of an empire, etc. In Greece, dates were often calculated based on the years of the Olympiad, so "in the year of the 322nd Olympiad," as an example.

The Western world tended to use the AUC calendar until the development of the calendar dated from the birth of Christ, in the Middle Ages, which is known as the Gregorian calendar. Scholars felt that it was unnecessary to date history from the founding of Rome, which, regardless of its Christian religion in the Middle Ages, was founded pagan, and also did not recognize the central event (in Christian philosophy) of history--the Incarnation of Jesus. Thus, they started to calculate years "in the year of Our Lord," or in Latin Anno domini--lots of people think "AD" means "after death," but that wouldn't make any sense, since Christ lived for about 33 years, and the years before his birth, which marked the end of "BC," and his death wouldn't fall into the system. That's just an aside, though, since Christ's death is dated around 33 AD.

Probably the most influential non-Western calendar is the Muslim calendar. For Muslims, the birth of Christ is not the central event that it is for Christians, and instead they use a major moment from the live of the Prophet Muhammad. The event that marks the beginning of the Islamic calendar is the "hijra," Arabic هجرة, or the "flight," when Muhammad and his followers fled from the city of Mecca. Thus, when Muhammad died ten years after the hijra, he died in 10 hijri, or 10 ه, which equates to 632 AD, or (if you want) 1375 AUC.

There are websites you can go to that help you convert these dates, like this one, although sadly lacking in AUC.

The Romans dated their years from the mythical founding of the city (ab urbe condita) in the year 753 BC. So Caesar in 70 BC would’ve thought of himself as living in the year 684 AUC, i.e in the 684th year after the founding of Rome.

That said, the most common way the Romans referred to specific years is by reference to the presiding consuls, e.g. “in the year when Aulus Hirtius and Gaius Pansa were consuls”. After the transition from Republic to Empire it became common for there to be up to 3 pairs of consuls in a single year (one ‘ordinary’ pair who were in charge at the start of the year, and up to two pairs of ‘suffect’ consuls who took over part way through the year), so in these cases the year was named after the ‘ordinary’ consuls who were in charge at the start of the year.

The situation in Classical Athens is similar, in that the year was named after one of the chief magistrates for that year, known as the ‘eponymous archon’. Olympiad dating (i.e. dating the year by the Olympic games) was also used as a standard across the various Greek city states.

Ancient Jews at the time of Jesus tended to use what is called Seleucid era dating, where the year is calculated from the year Seleucus I Nicator captured Babylon and began his reign over the Near East (311 BC). There is also some evidence in coinage that during the first Jewish revolt against the Romans (late 60s AD) Jews dated the year from the start of the revolt (e.g “in the 2nd year of the revolt”).

The current Hebrew calendar, developed in the Medieval period, uses anno mundi (“in the year of the world”) dating, counting the years from creation in 3761 BC.

TL;DR: there was no consistent ancient system for counting years. Ancient peoples tended to date their years from a significant event in the past and/or by reference to the presiding ruler, but systems varied greatly depending on time period and location.