When Pope Pius XI instituted the feast of Christ the King in 1925 he specified for it the last Sunday of October; Paul VI moved to the last Sunday of the liturgical year.
In the traditional Benedictine calendar however, this Sunday is indeed the feast of Christ the King.
The Kingship of Christ
In Quas primas, Pope Pius XI explained the basis for the feast:
"...these manifold evils in the world were due to the fact that the majority of men had thrust Jesus Christ and his holy law out of their lives; that these had no place either in private affairs or in politics: and we said further, that as long as individuals and states refused to submit to the rule of our Savior, there would be no really hopeful prospect of a lasting peace among nations. Men must look for the peace of Christ in the Kingdom of Christ; and that We promised to do as far as lay in Our power. In the Kingdom of Christ, that is, it seemed to Us that peace could not be more effectually restored nor fixed upon a firmer basis than through the restoration of the Empire of Our Lord. We were led in the meantime to indulge the hope of a brighter future at the sight of a more widespread and keener interest evinced in Christ and his Church, the one Source of Salvation, a sign that men who had formerly spurned the rule of our Redeemer and had exiled themselves from his kingdom were preparing, and even hastening, to return to the duty of obedience....
It has long been a common custom to give to Christ the metaphorical title of "King," because of the high degree of perfection whereby he excels all creatures. So he is said to reign "in the hearts of men," both by reason of the keenness of his intellect and the extent of his knowledge, and also because he is very truth, and it is from him that truth must be obediently received by all mankind. He reigns, too, in the wills of men, for in him the human will was perfectly and entirely obedient to the Holy Will of God, and further by his grace and inspiration he so subjects our free-will as to incite us to the most noble endeavors. He is King of hearts, too, by reason of his "charity which exceedeth all knowledge." And his mercy and kindness which draw all men to him, for never has it been known, nor will it ever be, that man be loved so much and so universally as Jesus Christ. But if we ponder this matter more deeply, we cannot but see that the title and the power of King belongs to Christ as man in the strict and proper sense too. For it is only as man that he may be said to have received from the Father "power and glory and a kingdom," since the Word of God, as consubstantial with the Father, has all things in common with him, and therefore has necessarily supreme and absolute dominion over all things created.
Do we not read throughout the Scriptures that Christ is the King?...Moreover, Christ himself speaks of his own kingly authority: in his last discourse, speaking of the rewards and punishments that will be the eternal lot of the just and the damned; in his reply to the Roman magistrate, who asked him publicly whether he were a king or not; after his resurrection, when giving to his Apostles the mission of teaching and baptizing all nations, he took the opportunity to call himself king, confirming the title publicly, and solemnly proclaimed that all power was given him in heaven and on earth. These words can only be taken to indicate the greatness of his power, the infinite extent of his kingdom...
Let Us explain briefly the nature and meaning of this lordship of Christ. It consists, We need scarcely say, in a threefold power which is essential to lordship. This is sufficiently clear from the scriptural testimony already adduced concerning the universal dominion of our Redeemer, and moreover it is a dogma of faith that Jesus Christ was given to man, not only as our Redeemer, but also as a law-giver, to whom obedience is due. Not only do the gospels tell us that he made laws, but they present him to us in the act of making them. Those who keep them show their love for their Divine Master, and he promises that they shall remain in his love. He claimed judicial power as received from his Father, when the Jews accused him of breaking the Sabbath by the miraculous cure of a sick man. "For neither doth the Father judge any man; but hath given all judgment to the Son." In this power is included the right of rewarding and punishing all men living, for this right is inseparable from that of judging. Executive power, too, belongs to Christ, for all must obey his commands; none may escape them, nor the sanctions he has imposed.
This kingdom is spiritual and is concerned with spiritual things..."
Reasons for celebrating the feast
If We ordain that the whole Catholic world shall revere Christ as King, We shall minister to the need of the present day, and at the same time provide an excellent remedy for the plague which now infects society....The empire of Christ over all nations was rejected. The right which the Church has from Christ himself, to teach mankind, to make laws, to govern peoples in all that pertains to their eternal salvation, that right was denied. Then gradually the religion of Christ came to be likened to false religions and to be placed ignominiously on the same level with them. It was then put under the power of the state and tolerated more or less at the whim of princes and rulers. Some men went even further, and wished to set up in the place of God's religion a natural religion consisting in some instinctive affection of the heart. There were even some nations who thought they could dispense with God, and that their religion should consist in impiety and the neglect of God. The rebellion of individuals and states against the authority of Christ has produced deplorable consequences...the seeds of discord sown far and wide; those bitter enmities and rivalries between nations, which still hinder so much the cause of peace; that insatiable greed which is so often hidden under a pretense of public spirit and patriotism, and gives rise to so many private quarrels; a blind and immoderate selfishness, making men seek nothing but their own comfort and advantage, and measure everything by these; no peace in the home, because men have forgotten or neglect their duty; the unity and stability of the family undermined; society in a word, shaken to its foundations and on the way to ruin.
We firmly hope, however, that the feast of the Kingship of Christ, which in future will be yearly observed, may hasten the return of society to our loving Savior. It would be the duty of Catholics to do all they can to bring about this happy result. Many of these, however, have neither the station in society nor the authority which should belong to those who bear the torch of truth. This state of things may perhaps be attributed to a certain slowness and timidity in good people, who are reluctant to engage in conflict or oppose but a weak resistance; thus the enemies of the Church become bolder in their attacks. But if the faithful were generally to understand that it behooves them ever to fight courageously under the banner of Christ their King, then, fired with apostolic zeal, they would strive to win over to their Lord those hearts that are bitter and estranged from him, and would valiantly defend his rights.
When we pay honor to the princely dignity of Christ, men will doubtless be reminded that the Church, founded by Christ as a perfect society, has a natural and inalienable right to perfect freedom and immunity from the power of the state; and that in fulfilling the task committed to her by God of teaching, ruling, and guiding to eternal bliss those who belong to the kingdom of Christ, she cannot be subject to any external power... Nations will be reminded by the annual celebration of this feast that not only private individuals but also rulers and princes are bound to give public honor and obedience to Christ."