An era is a continuous series of years measured by the occurrence of a significant event, whether legendary or historical. The earliest days of an era are called the epoch.
⦁ Olympiads: The Olympic Games were held in ancient Greece every four years, having initially been held on the first full moon following a summer solstice. The era of recording years based on when the Olympics was initiated in the 4th century BC, with dates recorded all the way back to the summer solstice of 1 July 776 BC. It ceased to see widespread usage with the abolition of the Olympic Games in 394 AD, with its last documented usage found in 12th century Germany.
⦁ Ab urbe condita: (years since the founding of Rome, abbreviation a. U. c.) According to tradition, the epoch of this era lies in the founding of Rome on 21 April, 753 BC. This era was compiled in 43 BC by Terentius Varro (* 116 BC - † 27 BC) based on official registers. It saw most of its use in the early Middle Ages, but also in later times by authors who sought to boast their classical education.
⦁ Era of the Martyrs (Diocletian era): this system has its origins in ancient Egypt, where the custom of counting years was according to whose reign it was. This custom was then carried over to the Roman emperors, who had ruled in Egypt as far back as 30 BC. The epoch of this era begins on 1 Thoth (a month on the Egyptian calendar) with the first year of Emperor Diocletian’s reign. This era was used from August 29, 284 AD until the Muslim conquest of Egypt in 640 AD.
⦁ Jewish age: According to tradition, Rabbi Hillel II, is considered to be the original creator of the new Jewish calendar. This age was calculated from what was believed to have been the creation of the world, using data and calculations derived from Biblical and Talmudic sources. The calendar was introduced circa 358~9 AD, with its epoch set to 7 October, 3761 BC.
⦁ Christian age: the most commonly used epoch today dates back to 525 AD, with a Roman abbot of Scythian origin named Dionysius Exiguus (* 470? AD - † 544? AD). Dionysius compiled Easter tables according to the Coptic system (see The First Council of Nicaea in 325) and found it inappropriate that years were counted according to the reign of Emperor Diocletian, the infamous persecutor of Christians. This was a motivating factor for why he instead opted to write “anni Domini nostri Jesu Christi” (in the year of our Lᴏʀᴅ, Jesus Christ). The year 247 of the Era of Martyrs (Diocletian era) was followed by year 532 of the Christian era, but nowhere did Dionysius ever explain how he came about to this number.
It is reasonable to assume that when in 525 AD (in fact, it was year 241 in the Era of Martyrs, but in the following text it’s best we continue to use our current era, as the hypothetical Dionysian reasoning is much more clear that way), while he was creating the tables for Easter, he noticed that in 563 AD Easter would occur on the 25th of March. According to tradition, Christ was resurrected on the 25th of March of an unspecified year, his death having transpired approximately 500 years prior to the time of Diocletian’s work. However, because the dates for Easter repeated after 532 years in the Julian calendar, he was able to easily calculate that the date of Christ's resurrection as 25 March, 33 AD. Based on this, he decided to establish Christmas as 33 AD, because of Christ’s 33rd birthday, and from there counting back to establish the first birthday of Jesus as having occurred in the year 1 AD (astronomical year 0).
Dionysius never mentioned what exactly the epoch of the Christian era was, the most likely reasoning being the appearance of 25 March - the day of the "incarnation of Christ" (a commonly held belief at the time was that the conception and resurrection of Jesus were on the same month and day). Or maybe Dionysius having set the epoch on the 1st of January as the beginning of the new epoch is true. Or perhaps it was set to the 25th of December, as an attempt to align the epoch with the birth of Christ. While I do not disqualify any of these as possibilities, it seems to me that the 25th of December is the most probable choice, because it makes it so that Christ was born on the first day of year 1; the fact Dionysius made it function that way makes all the stronger the case that he had intended it to work that way.
⦁ Byzantine calendar: constructed on the basis of chronological considerations in the Bible, as well as an effort to reconcile each year with a system that used Julian leap years, the Byzantine calendar’s usage dates back to the 7th century AD. It was used to facilitate the cyclical calculations of Easter as well as to assert the claim that its epoch begins with the creation of the entire world on 1 September, 5509 BC. This means that Year 1 of the Byzantine era ran from 1 September, 5509 BC to 31 August, 5508 BC. In effect, the dates between January and August never existed in 5509, only September through December did. This calendar garnered wide-scale use in the Eastern Orthodox regions, but slowly faded away after the fall of the Byzantine Empire, and then went on to have a second life of continuous use in the Balkans and Russia all the way up to the 18th century.