It seems unrealistic for the current system to have been used very early on. Was it used when emperor Constantine made Christianity Rome's official religion. Or was it much later, maybe the medieval period?
What you're asking about is the calendar era called Anno Domini (A.D. - "In the year of our Lord"). This is the current calendar era used in the West. There are, of course, other calendar eras with which are a familiar, but we might not use regularly. Two common ones would be the Jewish and Muslim calendars, both of which are lunar. The Jewish calendar uses a calendar era dating to the beginning of time (anno mundi), while the Muslim calendar uses a calendar era dating to the first Hijra (anno hegirae), the move of Muhammad and his followers to Medina from Mecca, traditionally dated as 622 CE. My personal favorite it the French Revolutionary Calendar. The revolutionaries just threw out the old Gregorian calendar and started anew. Alas, it did not catch on.
Regarding Anno Domini, it was first invented in the early Medieval period. The specific change is traditionally dated to Dionysius Exiguus's creation of the concept in 525 in Roman/Byzantine Syria*. He invented it to replace the then reigning Diocletian Era, which was pegged to the year 284, the year Diocletian began his reign. That system, while used in Syria, was not used in the West. In the West, after the fall of Rome, the dominant systems would have been Anno Mundi (likely in line with the Jewish calendar, but I can't say for sure) and regnal dating (i.e. the 4th year of the Reign of King Bvaze).
Anno Domini didn't catch on in Europe more generally until the Carolingian Renaissance, so 9th century. It starts to show up in the sources more often then, although I think Bede also uses it, and he's mid 8th century. And we still use it today.
EDIT: viceroylitton is correct. Dionysius Exiguus lived in Rome, but was from Roman Syria.
Originally the concept of giving the year in relation to Jesus' birth was formulated by Dionysius Exiguus, a monk working in Rome, in 525, he was working on a new Easter table when the idea occured to him that the years should really be based on jesus's birth and not the foundation of Rome as was common practice at the time. However it wasn't until the 800's until it becomes widely used, although Bede used A.D. for his Ecclesiastical History which was finished in 731, it appears that it was in fact the Carolingians and Emperor Charlemagne who made the use widespread on mainland Europe after they converted to the system Source: Dick Teresi, Zero, the atlantic, july 1997 http://www.theatlantic.com/magazine/archive/1997/07/zero/376900/
hi! your suspicions are correct: the Gregorian Calendar that most of us use today was introduced in 1582, well after the days of the Roman Empire. You'll find lots of info on what led up to this calendar, as well as some alternatives in the FAQ (including a nice overview in an AskHistorians Podcast on the topic)