When modern society speaks about the end of marriage, it usually means a lot of money paid to attorneys and a trip to the divorce court. In philosophical vocabulary, however, an end (Gk: telos, Lat: finis) refers to the final purpose of a thing, the goal towards which it is directed. When we look at the purposefulness of things, we are studying them under their teleological aspect. Teleology considers things in light of their final end, their ultimate purpose. The end of an acorn is to turn into an oak tree; if we anthropomorphize this reality, we could say the acorn "wants" to become a tree; in other words, the development of into a fully mature tree is the outcome the acorn tends toward by its very nature. Sacraments, too, have teleological ends; the end of the Sacrament of Penance is restoration of sinners to friendship with God through the forgiveness of their sins.
Marriage, of course, is one of the seven sacraments of the Church, and as such, has its own ends proper to it. In 1930, Pope Pius XI's groundbreaking encyclical Casti Connubii defined three ends of marriage: procreation, conjugal faith (later sub-defined to consist of "the good of the spouses" or "mutual aid", mutuum adiutorium and the "remedy of concupiscence") and the sacrament.  Because marriage, unlike the other sacraments, is a natural institution that was elevated to become a sacrament, it has ends which are natural as well as supernatural. The supernatural end of matrimony (the sacrament) is the indissoluble bond that arises as in every sacramental marriage and the graces that flow from that bond. But here we are concerned primarily with the natural ends of marriage, procreation and conjugal fidelity, because it is these ends that are most frequently called into question by moderns who wish to deny the Church's traditional teaching on marriage.
The identification of the ends of marriage was nothing new; in defining them in Casti Connubii, Pius XI basically takes as his reference point the teaching of St. Augustine. What was in question, however, was the relationship between the various ends of marriage, especially regarding procreation. The 1930 Lambeth Conference had legitimized the use of birth control for the Anglican Church, thus sundering the traditional unity of the two natural ends of marriage. Pius XI wrote Casti Connubii in response to this conference, and clearly taught that procreation and conjugal fidelity are inseparably united as ends of matrimony; furthermore, he clearly subordinated conjugal fidelity to procreation, stating that among the goods of marriage, "the child holds the first place" . This was in direct opposition to the Anglicans, who saw procreation as a salutary "side-effect" of matrimony but regarded the good of the spouses as the ultimate end.
The 1944 Ruling of the Sacred Roman Rota
Nevertheless, the relationship between the ends of marriage was still questioned or denied by many, prompting the Sacra Romana Rota to issue a judgment on the matter in 1944. The Roman Rota is the highest appellate court in the Catholic Church; its decisions are binding on both the Latin Rite and those of other rites in communion with Rome. The judgment was necessary because these dissenters were denying procreation as the primary end of marriage . The ruling of the Rota is not a Magisterial document per se, but it was inserted into the Acta Apostolica Sedis and has been used as a reference in other Magisterial documents, so it does enjoy a degree of authority.
The Rota ruling serves to clarify four main points:
I) The meaning of the term finis
II) The distinction between the ends of marriage as an institution and the individual ends of the spouses and the relation between these ends
III) The Church's understanding of each end
IV) The relation of each end to the marital institution itself
Drawing on a wealth of Catholic theological and canonical tradition, the Rota stated that there is an objective end of marriage and a subjective end. The word "end" utilized by the Church (finis) means a good that is to be obtained through any action, both with regards to the nature of the act itself, as well as the intention of the agents. Take the act of eating a dinner with friends. By the nature of the act itself, the finis of the act of eating is to satisfy the desire of hunger and procure the strength necessary for survival. That is the end as "intended" by nature. But eating with friends has another finis, one willed directly by the agent: in this case, spending time with companions and enjoying the social goods that derive from dining together. So we have two distinct sets of ends, one proper to the act itself, one willed by the specific agent. Yet both ends find their terminus or fulfillment in the same act.
Finis Operis and Finis Operantis
When discussing this point regarding marriage, the Rota employs the terms finis operis and finis operantis. The finis operis is the good to which marriage tends by its very nature objetively, by the design of the Creator, and which is intrinsic to the institution itself. Following the teaching of Pius XI, the finis operis of matrimony is procreation; this is what marriage tends to by the very nature of the conjugal act. Because the institution of marriage is a natural institution ordained by God and inscribed in human nature, it has certain ends towards which it tends by its very nature and which are not within the authority of man to transgress. This is a reaffirmation of what Pius XI had written in Casti Connubii:
"Christian doctrine establishes, and the light of human reason makes it most clear, that private individuals have no other power over the members of their bodies than that which pertains to their natural ends; and they are not free to destroy or mutilate their members, or in any other way render themselves unfit for their natural functions, except when no other provision can be made for the good of the whole body." 
Men are called to freely cooperate with God's plan by participating in marriage in loving acceptance of the ends intrinsic to the institution. Man can cooperate in pursuing the natural ends of marriage, but he has no authority to transgress the boundaries God has set. Man is a cooperator, but he does not have complete autonomy. Pope Pius XII said that man "is not the absolute master of himself, of his body, or his soul. He cannot, therefore, freely dispose of himself as he pleases." He is "bound to the immanent teleology laid down by nature" .
The finis operantis, on the other hand, concerns the particular goods willed by the spouses when they enter into matrimony. Because spouses are free moral agents, they must have some good in mind when they enter into marriage. These goods are as variable as are human beings and their motivations in marrying, but they can be summed up in the concept of happiness. Whatever the specific finis operantis in a given case, ultimately two persons enter into marriage because they believe they will be happier married than not. Finis operantis has to do with the intentions of the subjects.
What is the relation between the finis operis and the finis operantis? In looking at everyday practice, it is clear that these ends may coincide, or they may not. When a Catholic man and woman, well formed in the Church's teaching, enter into marriage, they may have a very lucid understanding of the natural ends of marriage, embrace these, and make them their own, thus synthesizing the finis operis and the finis operantis beautifully together, in such a way that the natural end and the end willed by the spouses are one and the same. Sometimes, however, the finis operantis may be different from the finis operis. We can think of a variety of reasons spouses may marry other than the raising up of a family. In these cases, the finis operis is simply something extra to the finis operantis in the intent of the spouses; in other words, the fact that the two ends do not synthesize does not mean the finis operis is necessarily negated or contradicted. The two ends can be divergent but not contradictory.
However, there are times when the finis operantis can be contradictory or incompatible with the finis operis. According to the Rota, "this happens every time that a person contracting matrimony has in mind a benefit or an end which is repugnant to one or all of the fines operis of matrimony" . If the couple attempts to engage in matrimony while intentionally willing ends incompatible with the finis operis, no marriage takes place due to a defect of consent because the natural ends to which marriage is structured by God are intentionally frustrated. This is why part of the act of consent in the marriage liturgy entails a specific assent to the finis operis of the institution: "Will you accept children lovingly from God and bring them up according to the law of Christ and his Church?" Though not incorporated as a vow, a similar reference to the finis operis is included in the pre-Vatican II rite, which speaks of marriage being instituted "for the continuation of the human race" and calls for the priest to bless the spouses with the prayer, "May you be blessed in your children, and may the love that you lavish on them be returned a hundredfold" and prays of the woman, " May she be rich in children" and "May she and her husband together see their children's children to the third and fourth generation and enjoy the long life that will fulfill their desires" . Failure to will the fines operis as specified in the rite, or to will to intentionally frustrate them, invalidates the marriage.
Thus, there are three possible relations the finis operis and the finis operantis can have to each other: They can coincide, they can be divergent but still compatible, or the finis operantis may be repugnant to the finis operis. Only in the last case is the contracting of a marriage impossible, for to enter into marriage, spouses must at least will the ends natural to the institution; if they intend the whole, they must intend its essential parts. This is one reason (among several) why a homosexual "marriage" can never be valid; the finis operis natural to the structure of marriage is procreation, and since two homosexuals can never will procreation as an end of their union, the finis operantis of a homosexual "marriage" would always be repugnant to the finis operis. This is certainly not the only argument against homosexual so-called "marriage" nor the strongest, but it is one objection and it is relevant to our discussion.
The Uniqueness of Matrimony
There are many types of human relationships, even friendships that are very intimate, but marriage is unique among all other relationships because it has a specific finis operis that it is ordered to by virtue of its institution by God. According to the Rota, marriage as a natural institution must have "a natural finis operis, one and indivisible, specifically proper and distinct from every other end" . It also states that in an institution where there are multiple ends, one must predominate, and others must be subordinated to it. In matrimony, the finis operis is the procreation and education of children, which is the primary end, and other ends must be subordinated to this one. It is important to note that the Rota rejects the notion that marriage has two primary ends; such a concept would undermine the vital relationship between the two ends and leave open the possibility of marriage without a procreative orientation, as we see today.
Additional Fines Operis
What are these additional ends? The procreation and education of children is the primary end, but within this primary end there are two secondary fines operis of matrimony: the good of the spouses and the remedy of concupiscence, which Pius XI highlights in Casti Connubii:
"For in matrimony as well as in the use of the matrimonial rights there are also secondary ends, such as mutual aid, the cultivating of mutual love, and the quieting of concupiscence which husband and wife are not forbidden to consider so long as they are subordinated to the primary end and so long as the intrinsic nature of the act is preserved." 
Though the secondary ends of the finis operis are logically distinct, they are interrelated; in fact, they cannot be realistically separated. For example, concupiscence can only be remedied through the lawful use of the sexual faculty within marriage. If the sexual act was being utilized outside of marriage, concupiscence would be inflamed, not remedied. It is only remedied when the sexual faculty is ordered towards its proper end. Likewise, the good of the spouses is related to procreation because it includes the common life that arises as a result of the unique marital relationship, which is predicated upon the raising up of a family. Even outside of marriage there can be relationships based on mutual help (as when Cardinal Ratzinger lived with his sister Maria), but the common life of spouses is different from the common life of friends or siblings because its relation to an internal primary end, which distinguishes it from other types of relationships by virtue of its ordination towards procreation. 
It is worth noting that the Rota and the Church in general do not equate procreation with biological reproduction. Procreation is an act of cooperation with God, and takes into consideration not only the physical creation of a child, but its conception in an act of willed self-giving, as well as the subsequent education and formation of the children. Procreation is a fundamentally holistic concept that concerns the entire human life of the child, not the mere act of biological reproduction. This is why artificial methods of reproduction that sunder the creation of new life from its familial context do not constitute licit methods of procreation in the eyes of the Church. The Church does not take a strict biologistic approach to the matter; this is also why actual conception of children is not necessary for a marriage to be ordered to procreation. Any marriage in which the natural ends are respected or not obstructed is ordered to procreation because the conjugal act is completed in accordance with the natural structure of marriage without opposing the finis operis. If this has been fulfilled, then the marriage is ordered to procreation, even if no actual procreation results from the union. In other words, the procreative activity of the couple does not require the actual conception of a child.
Because marriage is a natural institution ordained by God, it has certain intrinsic ends proper to it (fines operis). These ends are distinct from the subjective ends that may be willed by the spouses. Spouses can cooperate with these ends to varying degrees, so long as their subjective intention is not repugnant to the natural ends of marriage. In those cases, no marriage takes place. In addition to procreation, the good of the spouses and the remedy of concupiscence are other natural ends of marriage, but they are interrelated to procreation and must be subordinated to it. By procreation, the Church understands not a merely biological reproduction, but a cooperation with God in bringing new human beings into existence in the context of a loving family, which includes their education and care.
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 Pope Pius XI, Casti Connubii, 10 (1930)
 ibid., 11
 The case is #36 of the Acta Apostolica Sedis for 1944
 Casti Connubii, 71
 Pope Pius XII, "Address to the First International Congress on the Histopathology of the Nervous System", Sept. 14, 1952
 Quoted in Donald P. Asci, The Conjugal Act as a Personal Act (Ignatius Press: San Francisco, 2002), 41
 See the text of the rite in English and Latin here: http://www.catholicliturgy.com/index.cfm/FuseAction/TextContents/TextIndex/12
 Asci, 41
 Casti Connubii, 59
 Asci, 42-43