Holy days of obligation are feast days on which Catholics are required to attend Mass and to avoid (to the extent that they are able) servile work. The observance of Holy Days of Obligation is part of the Sunday Duty, the first of the Precepts of the Church.
There are currently ten Holy Days of Obligation in the Latin Rite of the Catholic Church and five in the Eastern Catholic Churches; in the United States, only six Holy Days of Obligation are observed.
WHAT IS AN OBLIGATION?
A lot of people misunderstand what it means to say that we are obligated to attend Mass on Sundays and Holy Days of Obligation. This isn't an arbitrary rule, but part of our general moral life—the need to do good and avoid evil. That is why the Catechism of the Catholic Church (Para. 2041) describes the obligations listed in the Precepts of the Church as "the very necessary minimum in the spirit of prayer and moral effort, in the growth in love of God and neighbor." These are things that, as Christians, we should want to do anyway; the Church uses the Precepts of the Church (of which the listing of Holy Days of Obligation is one) simply as a way to remind us of our need to grow in holiness.
WHEN ARE HOLY DAYS?
January 1st - Solemnity of Mary
Ascension - Celebrated 40 days after Easter usually on Sunday
August 15th - Feast of the Assumption
November 1st - Feast of All Saints
December 8th - Feast of the Immaculate Conception
December 25th - Christmas Day