The Wednesday, Friday, and Saturday after Ash Wednesday
The Wednesday, Friday, and Saturday after Pentecost
The Wednesday, Friday, and Saturday after the Feast of the Exaltation of the Holy Cross (Sept. 14)
The Wednesday, Friday, and Saturday after the feast of St. Lucy, which is Dec. 13
Ember days appear to be a very early Christian practice, first attested to as part of apostolic tradition by Pope Leo I in the fifth century. The purpose of their introduction, according to The Catholic Encyclopedia, was to thank God for the gifts of nature (hence their tie to the natural seasons), especially the crops used to make bread and wine for the Eucharist; to teach people to make use of those gifts in moderation; and to assist the needy.
Ember days also served as a response to the pagan festivals of Rome; the days encouraged Christians to counter the excesses and debauchery of those festivals by, instead, fasting and praying. At first the Church in Rome had fasts in June, September, and December, but the exact days were not fixed. The first record of the fasts for all four seasons being decreed comes in the writing of Pope Gelasius at the end of the fifth century.
After Gelasius, the practice spread beyond Rome. Gelasius also started the practice of permitting the conferring of ordinations on ember Saturdays, which were formerly given only at Easter, The Catholic Encyclopedia records. That tradition of holding ordinations of ember Saturdays also continues today, and in addition, ember days have traditionally been days of prayer for vocations.
The observance of ember days was later prescribed for the entire Latin Church by Pope Gregory VII (1073–1085). So, ember days were a big part of Catholic life for quite a while.
Though canon law no longer requires the observance of fasting and abstinence on ember days, they remain important for Catholics in many countries, and their continued observance by Catholics everywhere is certainly not discouraged. And in fact, some bishops in the United States have explicitly encouraged their Catholics to observe ember days and pray for specific intentions.