The Catholic Week and Month

Week

Fridays are penitential days and Catholics are to keep in mind Christ's suffering and to sacrifice something for the sake of penance and discipline. Catholics fulfill this duty by abstaining from meat and making other penances.

Saturdays are, traditionally, the days Catholics go to Confession in preparation for receiving the Eucharist on Sundays, but of course you can go to Confession on any day..

Sundays are, of course, the day for renewing Christ's once and for all Sacrifice during the Sacrifice of the Holy Mass. Because Christ rose from His tomb on Sunday, Sabbath was changed from Saturday to Sundays, or "the Lord's Day." On this day we fulfill God's Third Commandment, to "remember the Sabbath day [which means "rest", not "Saturday"], to keep it holy." We refrain from unnecessary servile work and fulfill our "Sunday Obligation" to attend Mass.

Month

Footnotes:
¹ Tuesdays are also the day for honoring the Holy Face and, unofficially, honoring St. Anthony of Padua. On the Tuesday after St. Anthony's death, the day his funeral cortege took his body to the church, many miracles took place, so this day is commemorated in his honor, often by praying a Novena to St. Anthony either for thirteen consecutive Tuesdays or perpetually.

A special Novena to St. Dominic, St. Anne and St. Martha is also made on 9 consecutive Tuesdays or on all Tuesdays by some Catholics.

² On Wednesdays, many Catholics make a special devotion to St. Joseph by going to Mass on the first Wednesdays of 9 consecutive months and offering their Communions in his honor and for the salvation of the dying..

³ On Thursdays, many Catholics make "Holy Hour," that is, they spend an hour in adoration of the Blessed Sacrament as an aspect of devotion to the Sacred Heart.

⁴ On Fridays, many Catholics make what is known as the "First Fridays Devotion" in honor of the Sacred Heart. This entails going to Mass and receiving Communion in reparation to the Sacred Heart on the first Friday of the month for 9 consecutive months.

⁵ On Saturdays, many Catholics make what is called the "First Saturdays Devotion" which entails going to Mass and receiving Communion for the first Saturday of the month for 5 consecutive months in reparation to the Immaculate Heart of Mary.   

Partial Text courtesy of Fisheaters

The Catholic Year Calendar-Related Customs

Below are links to pages that describe customs and devotions related to the liturgical year.
 Dates in italics are movable.

The Catholic year (the "liturgical year") is made special by cycles of celebrations commemorating the lives of Jesus and His mother, the angels, and the legion of Saints who modeled lives of sanctity. Below are 25 Feasts and times, in chronological order, that demonstrate how the liturgical year is a reliving of the life of Christ:

Every single year, aware Catholics "re-live" the Gospel, from Christ's Incarnation and Birth to His Ascension and Heavenly reign. In Spring He enters the world by coming to rest in Mary's immaculate womb; nine months later, in Winter, He is born, circumcised, and given a Name. He is raised in the Holy Family, and meets His cousin, John. He goes into the Desert and we go with Him during our Lenten Season. Then follow His Passion and Agony, which are soon vanquished by His Resurrection, His Ascension, and the Pentecost. Now He reigns -- and forever, and we await His Second Coming as we prepare to celebrate again His First Coming. Then the cycle begins again, like a wheel that's been spinning for two millennia. The Catholic who is aware of this wheel is necessarily aware of Christ; the Catholic who also celebrates the Feasts well and practices the traditions of the Church lives intimately with Him.

All of the Church's Feasts¹ fall into one of the 2 main "liturgical cycles" made of 7 "liturgical seasons." Each of the Seasons has an associated mood, its own "feeling in the air," its own scents and ornaments. There is even for each Season an associated color which will be reflected in the priests' vestments and liturgical art, church decoration, and so on (though on certain Holy Days within a particular season, that Day's color will take precedence over the season's color).

"Overlaid" on this grid of Seasons are two sets of dates: the Proper of Saints (also called the "Sanctoral cycle") and the Proper of Seasons (also called the "Temporal cycle"). The Proper of Saints are Feast Days which are not movable, that is, they fall on the same date each year. The Proper of Seasons are those Sundays and other Feasts of the year, whose dates of celebration depend on the dates of Easter Sunday and Advent Sunday and are, therefore, movable (they change each year).

To determine the dates of the Proper of Seasons:

Then refer to the Temporal Cycle to fill in any movable Feasts whose dates depend on the date of Easter or Advent Sunday as determined above. The only things left to do are: 

Now, each of the Sundays of a Season has its own "Propers" -- prayers that are specific to that day in the liturgy (the Divine Office and the Mass). Each of the Feasts in the Proper of Saints will also have its own Propers. So, because the Feasts in the Proper of Saints and the Proper of the Seasons can sometime overlap with two Feasts falling on the same day, all Feasts are ranked according to their importance. The higher ranking Feast will be the one celebrated.

Feasts fall into one of a few categories, in descending order of precedence ²:
 

In older Missals, the Feasts are ranked thus:

When two Feasts of the same rank fall on the same day, they are ranked further by whether they relate to (in descending order of preference):

A "feria" (the word means "free day") is a "weekday" -- that is, a day that is neither a Sunday nor any other Feast.

Majority of text courtesy of Fisheaters