An Explanation of the Liturgical Year

Unlike pagan religions which see time as an endless cycle, Christians see time as being linear; it has a beginning and will have an end. Within Christianity's linear, "big picture" sense of time, though, the passing of hours is experienced as cycles of meditations on holy things. Think of a spiral -- of a circle of time moving ever forward toward His Coming -- and you will have a sense of "Catholic time."

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 modelled 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:

Advent -He is coming
Nativity - He comes
Circumcision - He follows Old Testament Law
Epiphany - He reveals Himself as God
Holy Family -He grows up in a human family
Candlemas - Simeon's prophecy
Septuagesima - We are in exile without Christ
Ash Wednesday - Without Christ, we are dust
Lent - Christ is in the Desert
Passion Sunday - Jews make plans to kill Jesus
7 Sorrows - Mary's suffers at what is to come
Palm Sunday - He triumphantly enters Jerusalem
Spy Wednesday - Jesus is betrayed by Judas
Maundy Thursday - He offers the first Holy Mass
Good Friday - He is put to death and fulfills Old Testament Law
Holy Saturday - He is in the tomb
Easter - He is risen
Ascension - He ascends into Heaven
Pentecost - He sends the Holy Ghost
Trinity Sunday - The Most Holy Trinity has been fully revealed
Assumption - Mary is assumed into Heaven & crowned Queen
Christ the King - We recognize Christ's Kingship now and forever
All Saints - We will triumph as have our heroic Saints
All Souls - We pray for those who are awaiting their triumph
Last Sunday in Time after Pentecost
Apocalypse. He will come to judge the world.

"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).
In other words, to imagine the liturgical year:

Imagine a regular, standard, everyday calendar.

Mentally overlay on that the Proper of the Saints, filling in each day of the regular calendar with the names of the Feasts for each day, the dates of which don't change -- e.g., January 21 will always be the Feast of St. Agnes, February 3 will always be the Feast of St. Blaise, etc. Then determine the dates of the Proper of Seasons and overlay that on top of the Proper of Saints.

To determine the dates of the Proper of Seasons:

Mark the Season of Easter:
First, we determine the date of Easter, which will be the first Sunday after the first full moon on or after March 21 (even if the full moon on or after March 21 falls on a Sunday, go to the Sunday after). The Vigil of this Feast marks the beginning of Eastertide. Mark the Season of Time after Pentecost:
Counting Easter as "one," count 9 Sundays forward from Easter and mark that Sunday as the beginning of Time After Pentecost. A Sunday of this Season is referred to as "(First, Second, Third, etc). Sunday after Pentecost."

Mark the Season of Septuagesima:
Counting Easter as "one," count 10 Sundays back from Easter and mark that day as the beginning of Septuagesima. The three Sundays of this Season are referred to, respectively, as Septuagesima Sunday, Sexagesima Sunday, and Quinquagesima Sunday.

Mark the Season of Lent:
Counting Septuagesima Sunday as "one," count 3 Sundays forward from Septuagesima Sunday, then go to the following Wednesday and mark that Wednesday as "Ash Wednesday," the beginning of Lent. A Sunday in this Season is referred to as "(First, Second, Third, etc). Sunday of Lent."

Mark the Season of Advent:
Then, starting with the date of Christmas (always December 25), we count back 4 Sundays to mark Advent Sunday (if Christmas is a Sunday, don't count it; count back 4 entire Sundays so that there are 4 Sundays in Advent). Another way to do this is to simply mark the Sunday closest to St. Andrew's Day (30 November). This date marks the beginning of Advent. A Sunday in this Season is referred to as "(First, Second, Third, etc.) Sunday of Advent."

Mark the Season of Christmas:
Mark the Vigil of December 25 as the beginning of Christmastide

Mark the Season of Time after Epiphany:
Mark January 14 as the beginning of Time After Epiphany. A Sunday of this Season is referred to as "(Second, Third, etc.) Sunday after Epiphany." Note, the first Sunday of this Season is the "Second Sunday after Epiphany," the "after Epiphany" referring to the Feast of the Epiphany, not to the Season.

Then refer to the Temporal Cycle page 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:
to mark the "Octaves":

Octaves are 8-day periods of observance, beginning with the Feast day itself. Not all Feasts have "Octaves"; only the most important ones do. So, starting with the Feast Day itself, counting it as "one," mark 8 days of the following Feasts as "Octaves": Christmas, Easter, and Pentecost. Then mark the octave before Christmas Eve as "The Golden Nights."

To mark Ember Days and Rogation Days:
the Wednesday, Friday, and Saturday after the Third Sunday of Advent (Gaudete Sunday) are the days of Advent Embertide
the Wednesday, Friday, and Saturday after the First Sunday of Lent are known as Lenten Embertide
the Wednesday, Friday, and Saturday after Pentecost Sunday make up Whit Embertide
the Wednesday, Friday, and Saturday after the Feast of the Exaltation of the Holy Cross (14 September). Note that these Ember Days must come a full week after the Holy Cross Day.

Mark the Major Rogation on April 25, and the Minor Rogation on the three days -- Monday, Tuesday, and Wednesday -- before Ascension Thursday to mark your cathedral's patronal Feast:
Mark the Feast of the patron Saint of your diocese's cathedral (e.g., if your cathedral is named "SS. Peter and Paul Cathedral," the priests of your diocese will celebrate 29 June, the Feast of SS. Peter and Paul, as a first class Feast

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:
1st Class
2nd Class
3rd Class
Commemoration

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):
Our Lord
Our Lady
the Holy Angels
St. John the Baptist, St. Joseph, St. Peter, St. Paul, the other Apostles
Martyrs
Other Saints

If a day is neither a Sunday nor commemorative of any other Feast, it is called a "feria" (the word means "free day").

Holy Days of Obligation

In addition to each Sunday, there are a handful of other Holy Days of Obligation on which we must attend Mass. These Holy Days differ from country to country. In the United States, these days are;

Circumcision
Ascension
Assumption
All Saints
Imm. Conception
Christmas

Other days a family might want to mark on their home calendars are:
the family's "Name Days"
the First Friday and First Saturday of each month for devotions to the Sacred Heart and to the Immaculate Heart respectively.
The anniversaries of loved ones' deaths so we remember to have Masses said for them, and light candles, fast, and pray for them.
The liturgical year is less confusing than it seems at first, but to follow along, you can do what most Catholics do and just pay attention to your parish bulletins and/or get a Catholic calendar each year.

Footnote: As our family is not Catholic but do follow the Liturgical Calendar in our home, we do not attend Mass on Holy Days of Obligation.  We do however, honor this day with a small service of our own here in our home.