What is the difference between a "feast" and a "solemnity" and why are some days "ordinary" and some days not?
The Church celebrates the special events that are important in the life of the Church. Also, we celebrate the lives of saints, who through their lives, bring to the minds of the faithful the call to holiness. Saints’ days are usually celebrated on the date of the saint’s death– their birth into eternal life. There are a few exceptions to this.
The difference between these celebrations are merely degrees of relative importance. Solemnities are the celebrations of greatest importance and begin on the prior evening with evening prayer in the Liturgy of the Hours. Several solemnities have their own Vigil Mass. On these days, both the Gloria and the Creed are recited during Holy Mass. Sundays and Holy Days of Obligation are always considered solemnities.
Feasts are of secondary importance in our liturgical calendar and are celebrated on a particular day. These feasts do not have a special evening prayer in the Liturgy of the Hours nor do they have their own special Vigil Mass the prior evening: an exception would be the feasts of the Lord which occur on Sundays in Ordinary Time and Sundays in the Christmas season. For example, the Feast of the Presentation of the Lord (February 2) has its own first vespers. On these days, the Gloria is recited but not the Creed. Examples of Solemnities are Easter, Christmas, All Saints Day, the Ascension, and Corpus Christi.
Even though you did not ask, there is also another celebration called a memorial, which is either obligatory or optional. Memorials commemorate a saint or saints. Obligatory memorials must be observed whereas optional memorials do not have to be observed. For example, the memorial of St. John Bosco (January 31) is obligatory while the memorial of St. Blase (February 3) is optional. Only the memorials of those saints who are of “universal significance” are observed by the whole Church and marked in the general liturgical calendar.
Particular churches, countries, or religious communities may also celebrate the memorials of other saints of “special significance” in accordance with their special devotions. For example, the memorial in honor of Saint Louis, King of France, in our own archdiocese is raised to the level of a “feast.”
The litugical calendar changes each year. You can see a list of the litugical calendar for 2016 at the following link:
http://www.usccb.org/about/divine-worship/liturgical-calendar/upload/2016cal.pdf and for the coming new year here: http://www.usccb.org/about/divine-worship/liturgical-calendar/upload/2017cal.pdf
With regards to "ordinary days" this is often a source of confusion. One meaning of the word "ordinary" is "usual, unimportant, not extraordinary". This is not the meaning of the word as it applies to the Church calendar.
Ordinary Time is called "ordinary" - not because it is unimportant - but simply because the weeks of Ordinary Time are numbered. The Latin word ordinalis, which refers to numbers in a series, (1st, 2nd, 3rd, .... etc.) stems from the Latin word ordo, from which we get the English word order. Thus, the numbered weeks of Ordinary Time in fact represent the ordered life of the Church—the period in which we live our lives neither in feasting (as in the Christmas and Easter seasons) or in more severe penance (as in Advent and Lent), but in watchfulness and expectation of the Second Coming of Christ.
We hope this helps.