Tradition obviously has it that Jesus was born in the year 1AD, but I can't image that as Jesus was born people decided they had better start counting the years straight away. Surely there must have been a point in time where it was decided that we should have a system for keeping track of the years, and they kind of back dated towards Jesus' birth. But is there any record of when this happened?
Just as a quick sidenote, this is mostly based on this earlier answer from myself. /u/Compieuter
The calender system that we use was started by Dionysius Exiguus (470-544), a Roman from what is now roughly Bulgaria. Dionysius lived in a world with many calenders, regnal calenders were popular but there were many more and the Roman empires and Christians had great difficulty on agreeing to use a single calender. A regnal calender is calender based on the years of rule from the monarch. In the Roman empire there wasn't really one accepted way of counting the years, they would name each year after the two consuls of that year. After a while, I think only in the first century BC, they would sometimes number them Ab Urbe Condita. AUC is from the founding of the city which was suposedly in 753 BC. But this way of counting the years was never that popular and within the Roman empire you also had Jews with their own calender and if I recall corectly many Greeks had a year system based on the Olympic games.
Roman emperor Diocletian at the end of the 3rd century was able to more or less standardise things a bit in the empire and in the time of Dionysius the most common calender system counted the years from the start of the Diocletians reign. Dionysius wasn't really a fan of this, because Diocletian was a pagan emperor that had even persecuted Christians during his reign. So Dionysius set out to set up a different calender in his new home, the city of Rome. Dionysius had moved from the Eastern Roman empire to go and work for the Pope/Patriarch in Rome which was part of the new Gothic kingdom. The Gothic kingdom and the Eastern Roman empire would fight a decades long war later, during Dionysius lifetime, but that is a story for another time.
Dionysius wanted a new calender without any reference to what in his mind was a brutal, pagan, christian killing emperor. So he had to find a new starting point and what better starting point for a pious monk to choose then the birth of his spiritual king in heaven, Jesus Christ. A quick sidenote here is that one of the main functions of calenders was to reliably predict the date of easter, because that was difficult to calculate. In this Dionysius used the Egyptian method developped in Alexandria but this is all quite complicated. Since I don't have my sources with me and it doesn't directly relate to your question I will skip that part.
Back to Jesus, Dionysius made this new calender with a better way to calculate easter and a new starting point. Picking the starting point wasn't so easy though as our source material for the life of Jesus is unfortunately vague, contradictory or absent. We don't have any contemporary sources attesting Jesus apart from the different books of the bible. This might seem strange but a lot of writings from antiquity were lost and Jesus was maybe only an apocalyptic preacher in a far away province who gathered a following of a dozen or so men and women that was only active for roughly three years. It's not surprising that we don't have any Roman documents on him. What we do have is the bible, specifically the books of Luke and Mathew. These two sources make three different refferences to the dating of Jesus' life:
Mathew 2:1 says that Jesus was born during the reign of King Herod (a vassal to the Romans) who died in what is now known as the year 4 BC.
Luke 2: 1-2 says that Jesus was born in the time of a census ordered by emperor Augustus (r. 27 BC - 14 AD) during the governourship of Quirinius in AD 6 and 7.
Luke 3: 1-2 and Luke 3:23 state that Jesus was baptised when he was roughly 30 years old in the fifteenth year of emperor Tiberius which coresponds to AD 28 and 29.
The problem with these three pieces is that they don't match. Recently there have been many internet articles arguing that Dionysius was dumb and that he made a mistake by picking 1 AD and that Jesus was actually born in 2 BC but there is now way to truelly know what was the actual date. The reasons for Dionysius picking the year 1 AD are also unknown but there are some theories rangeing from him just compromising between the different dates and Dionysius taking into acount astrlogical events like the lining up of the planets. He might have made a mistake but picking any date between 4 BC and AD 7 would have just as much legitimacy.
Anyway we can say with relative certainty that a person that was later called Jesus and about which they wrote the new testament was born in roughly AD 1. Why didn't he start with zero ? Well the concept of the number zero did exist and was maybe known to some people in the Roman empire but it took roughly five more centuries before the use of the number zero would become even a bit popular in Christian Europe. However, even if Dionysius did have a concept of the number zero then I don't think that he would have used it in his calender. As his calender like many other calenders before and after him was also in a way a regnal calender. Regnal calenders usually counted the years of the ruler, for example we are currently in the second year of the American President Trump . Dionysius calander counted the years of Jesus as his king in heaven, so we are currently in the 2018th year of Jesus as the (Christian) king in heaven.
The exact date of Jesus birth is also another mystery that not everyone agrees on so Dionysius is a bit vague about his starting point and decided to stick with the system that was in place, which was the Julian calender with 12 months and that started roughly in the middle of winter on January first. Despite this effort from Dionysius to fit into the existing calenders it took a very long timefor his calnder to become the standard in Catholic Europe. The Northumbrian Venerable Bede (d. 735) and Pope Nicolas II (d. 1061) played important parts in it's popularisation but for many parts of Europe it took until the 16th or 17th century for this calender to fully replace the local regnal calenders.
Sources:
Declerc G., Anno Domini. The Origins of the Christian Era. (Turnhout, 2000).
And for the planetary alignment theory check out:
Rothwangl S., ‘The Scythian Dionysius Exiguus and His Invention of Anno Domini’ (2016).
There's a really in-depth and well explained answer to a similar question by /u/sunagainstgold here - hope that helps!