The Roman Church is governed by laws, formulated as the Code of Canon Law. Canon 1246 discusses and lists the holy days of obligation.
The Lord’s Day (Sunday), “on which the paschal mystery is celebrated,” is to be observed as the “primary holy day of obligation.” For feasts that occur on days other than Sunday, there are additional holy days of obligation: Christmas, Epiphany, the Ascension, Body and Blood of Christ (Corpus Christi), Mary Mother of God (Jan. 1), Immaculate Conception, Assumption, St. Joseph, SS Peter and Paul, and All Saints.
Section 2 of Canon 1246 states that the “Episcopal Conference” (in our case here in the United States, the USCCB) may “suppress” certain holy days of obligation, or transfer them to a Sunday, with the prior approval of the Apostolic See (i.e., the Pope).
If you examine the liturgical calendar for the United States, you will find that Epiphany occurs on Sunday, January 5 this year. However, Epiphany is actually always January 6, the twelfth day after Christmas.
Thus, in the case of Epiphany, the American bishops have seen fit to transfer, rather than suppress, the Feast of the Epiphany to the nearest Sunday (January 5). It remains a holy day of obligation, in the sense that Sundays are always the primary holy days of obligation.
By comparison, the feast which celebrates the spouse of the Blessed Mother, St. Joseph (March 19), has been “suppressed” by the American bishops, who most recently decreed which holy days are observed in the United States in 1991.
The Feast of St. Joseph is big in certain countries, like Italy (where St. Joseph is the national patron), Spain, etc., and often takes the place of our secular “Father’s Day” here. Since at least part of the month of March is usually included in the 40 days of Lent, the day (as a “solemnity”, or highest ranking feast day) permits one to dispense with doing penance, as one would be permitted to do on any Sunday in Lent. Likewise, St. Patrick’s Day (March 17), while a major drinking holiday here and a time for American meat producers to offload all their extra brisket, is actually a holy day of obligation in Ireland, where the Irish bishops have seen fit to celebrate St. Patrick as the nation’s patron.
As the saying goes, if you fast when the Church feasts, you fast alone.