Anyone who has been Catholic for at least a few years is probably aware that, on occasion, holy days of obligation conflict. For instance, in 2013, December 8 (the Feast of the Immaculate Conception) fell on a Sunday, specifically, the Second Sunday of Advent. Since, in the reformed calendar, the Sundays of Advent are of superior rank to all other solemnities and are immoveable, the observance of the Feast of the Immaculate Conception in the ordinary form calendar was transferred to Monday, and no mention of it was made on its proper Sunday. This had to be done because, in the ordinary form, it is not possible to honor two distinct occasions or saints in the same Mass: we can have the Sunday of Advent, or we can have the Immaculate Conception, but we can’t have both at the same time; and since the Immaculate Conception is less important than the Sundays of Advent, it got the boot to Monday. That’s at least better than ignoring the occasion entirely, but it’s pretty regrettable that no mention of the Immaculate Conception is made on the day proper to the Immaculate Conception, isn’t it? This is doubly true as comparatively few people are willing to attend Mass on Monday, even in places where the opportunity is available.
The traditional Latin Mass (as well as at least one Eastern liturgy, that of the Byzantines) gets around the issue of conflicting feasts in a way unfamiliar to most Catholics, called commemoration. When two feasts fall on the same day, the Mass may be celebrated according to the rubrics for the higher-ranked feast, with the lesser feast commemorated. This means that certain of the lesser feast’s propers (specifically, the Collect, Secret, and Postcommunion) are prayed right alongside those of the superior feast. Here’s an example of commemorating the Collect in the traditional Latin Mass, from December 8, 2013 (in the traditional calendar, the Immaculate Conception enjoys precedence, so the Second Sunday of Advent was commemorated):
V. Dóminus vobíscum. (The Lord be with you.) R. Et cum spiritu tuo. (And with your spirit.) Orémus. (Let us pray.) Deus, qui per immaculátam Vírginis Conceptiónem dignum Fílio tuo habitáculum præparásti: quaesumus; ut, qui ex morte eiúsdem Filii tui prævísa eam ab omni labe præservásti, nos quoque mundos eius intercessióne ad te perveníre concédas. Per eundem Dominum nostrum Iesum Christum filium tuum, qui tecum vivit et regnat in unitate Spiritus Sancti, Deus, per omnia saecula saeculorum. (O God, Who by the Immaculate Conception of the Virgin, prepared a worthy dwelling for Your Son, and Who, by Your Son’s death, foreseen by You, preserved her from all taint, grant, we beseech You, through her intercession, that we too may come to You unstained by sin. Through Jesus Christ, thy Son our Lord, Who liveth and reigneth with thee, in the unity of the Holy Ghost, ever one God, world without end.) R. Amen. Orémus. (Let us pray). Excita, Dómine, corda nostra ad præparándas Unigéniti tui vias: ut, per eius advéntum, purificátis tibi méntibus servíre mereámur: qui tecum vivit et regnat in unitate Spiritus Sancti Deus per omnia saecula saeculorum. (Stir up our hearts, O Lord, to prepare the ways of Your only-begotten Son, so that through His coming we may be able to serve You with purified minds. Who livest and reignest with God the Father, in the unity of the Holy Spirit, one God, world without end.) R. Amen.
Neat, huh? In older times, commemoration went even further: the Gospel of the lesser feast was proclaimed in place of the Johannine Prologue at the Last Gospel, a feature suppressed under the reign of Pius XII.