Beginning of Advent

When Did the Celebration of Advent Begin?
The recognition of the liturgical season began in about the 5th or 6th century in the Latin Church and there is no record of Advent prior to the 8th century in the Eastern Church (source).  There is some mention at a synod held in Saragossa in 380 which disallowed absence from Mass between December 17 and Epiphany decreed in the fourth canon from that synod (ibid).  So it is possible this seasonal celebration began in the 4th century shortly after Roman persecution of the Church had ended and Catholics could openly celebrate and publicly announce liturgical rites and seasons.

From the Aquinas and More website:
The exact time when the season of Advent came to be celebrated is not precisely known. Of course, it was not in practice before the celebration of the Nativity and Christmastide began; the earliest evidence shows that the feast of the Nativity of Our Lord was established within the later part of the 4^th century. There are homilies from the 5^th century that discuss preparation in a general sense, but do not indicate an official liturgical season. A Synod held in 590 established that Mondays, Wednesdays, and Fridays from November 11^th until the Nativitywould be offered according to the Lenten rite. This and other traditions, such as fasting, show that the period of time now established as the Advent season was more penitential (similar to Lent) than the liturgical season as we know it today.

A collection of homilies from Pope St. Gregory the Great (whose papacy was from 590-604) included a sermon for the second Sunday of Advent, and by 650 Spain was celebrating the Sundays (five at the time) of Advent. So it seems the liturgical season was established around the latter part of the 6^th century and first half of the 7^th century. For the next couple of centuries, Advent was celebrated for five Sundays; Pope Gregory VII, who was pope from 1073-85, reduced the number to four Sundays.  (Source).