Marking the Changes of the Liturgical Seasons:
Before the revision of the Church's liturgical calendar in 1969 (coinciding with the adoption of the Mass of Paul VI), Ember Days were celebrated four times each year. They were tied to the changing of the seasons, but also to the liturgical cycles of the Church. As Fr. John Hardon, S.J., writes in his indispensable Modern Catholic Dictionary, "They were the Wednesday, Friday, and Saturday after St. Lucy (December 13), the First Sunday of Lent, Pentecost, and the feast of the Holy Cross (September 14)."
The Roman Origin of Ember Days:
It's common to claim that the dates of important Christian feasts (such as Christmas) were set to compete with or replace certain pagan festivals, even though the best scholarship indicates otherwise.
In the case of the Ember Days, however, it's true. As the Catholic Encyclopedia notes:
The Romans were originally given to agriculture, and their native gods belonged to the same class. At the beginning of the time for seeding and harvesting religious ceremonies were performed to implore the help of their deities: in June for a bountiful harvest, in September for a rich vintage, and in December for the seeding.
Keep the Best; Discard the Rest:
The Ember Days are a perfect example of how the Church (in the words of the Catholic Encyclopedia) "has always tried to sanctify any practices which could be utilized for a good purpose." The adoption of the Ember Days wasn't an attempt to displace Roman paganism so much as it was a way to avoid disrupting the lives of Roman converts to Christianity. The pagan practice, though directed at false gods, was praiseworthy; all that was necessary was to transfer the supplications to the true God of Christianity.
An Ancient Practice:
The adoption of Ember Days by Christians happened so early that Pope Leo the Great (440-61) considered the Ember Days (with the exception of the one in the spring) to have been instituted by the Apostles. By the time of Pope Gelasius II (492-96), the fourth set of Ember Days had been instituted. Originally celebrated only by the Church in Rome, they spread throughout the West (but not the East), starting in the fifth century.
The Origin of the Word:
The origin of the word "ember" in "Ember Days" is not obvious, not even to those who know Latin. According to the Catholic Encyclopedia, "Ember" is a corruption (or we might say, a contraction) of the Latin phrase Quatuor Tempora, which simply means "four times," since the Ember Days are celebrated four times per year.
With the revision of the liturgical calendar in 1969, the Vatican left the celebration of Ember Days up to the discretion of each national conference of bishops. They're commonly celebrated in Europe, particularly in rural areas.
In the United States, the bishops' conference has decided not to celebrate them, but individual Catholics can and many traditional Catholics still do, because it's a nice way to focus our minds on the changing of the liturgical seasons and the seasons of the year. The Ember Days that fall during Lent and Advent are especially useful to remind children of the reasons for those seasons.
Fasting and Abstinence:
The Ember Days are celebrated with fasting (no food between meals) and half-abstinence, meaning that meat is allowed at one meal per day. (If you observe the traditional Friday abstinence from meat, then you would observe complete abstinence on an Ember Friday.)
As always, such fasting and abstinence has a greater purpose. As the Catholic Encyclopedia notes, through these activities, and through prayer, we use the Ember Days to "thank God for the gifts of nature, . . . teach men to make use of them in moderation, and . . . assist the needy."