The Pope is the visible head of the entire Church, and Christ is the invisible Head.

Pope's visible headship

First Vatican Council, Pastor Æternus ch. 4, reiterating the Council of Florence*:

the Roman Pontiff is the true vicar of Christ, and the head of the whole Church, and the father and teacher of all Christians; and that to him in blessed Peter was delivered by our Lord Jesus Christ the full power of feeding, ruling, and governing the whole Church.

_{*The Council of Florence defined the dogma (Denzinger 694):}

^{We likewise define that the holy Apostolic See, and the Roman Pontiff, hold the primacy throughout the entire world; and that the Roman Pontiff himself is the successor of blessed Peter, the chief of the Apostles, and the true vicar of Christ, and that he is the head of the entire Church, and the father and teacher of all Christians; and that full power was given to him in blessed Peter by our Lord Jesus Christ, to feed, rule, and govern the universal Church; just as is contained in the acts of the ecumenical Councils and in the sacred canons.}

ibid. ch. 1:

If any one, therefore, shall say that blessed Peter the Apostle was not appointed the Prince of all the Apostles and the visible Head of the whole Church Militant; or that the same directly and immediately received from the same our Lord Jesus Christ a primacy of honor only, and not of true and proper jurisdiction: let him be anathema.

Christ's invisible Headship

In Summa Theologica III q. 8 a. 1 ("Whether Christ is the Head of the Church?") co., St. Thomas Aquinas shows that order, perfection, and power are essential attributes of being a head:

1. Order, indeed; for the head is the first part of man, beginning from the higher part; and hence it is that every principle is usually called a head according to Ezech. 16:25: "At every head of the way, thou hast set up a sign of thy prostitution"
2. Perfection, inasmuch as in the head dwell all the senses, both interior and exterior, whereas in the other members there is only touch, and hence it is said (Is. 9:15): "The aged and honorable, he is the head"
3. Power, because the power and movement of the other members, together with the direction of them in their acts, is from the head, by reason of the sensitive and motive power there ruling; hence the ruler is called the head of a people, according to 1 Kgs. 15:17: "When thou wast a little one in thy own eyes, wast thou not made the head of the tribes of Israel?"

Then he shows these attributes apply to Christ:

Now these three things belong spiritually to Christ.

1. [Order,] on account of His nearness to God His grace is the highest and first, though not in time, since all have received grace on account of His grace, according to Rm. 8:29: "For whom He foreknew, He also predestinated to be made conformable to the image of His Son; that He might be the first-born amongst many brethren."

2. He had perfection as regards the fulness of all graces, according to Jn. 1:14, "We saw Him [Vulg.: 'His glory']… full of grace and truth"

3. He has the power of bestowing grace on all the members of the Church, according to Jn. 1:16: "Of His fulness we have all received."

And thus it is plain that Christ is fittingly called the Head of the Church.

Invisible vs. visible headship

In ibid. a. 6 ("Whether it is proper to Christ to be Head of the Church?") co., St. Thomas explains that a

head influences the other members in two ways.

1. by a certain intrinsic influence, inasmuch as motive and sensitive force flow from the head to the other members;
2. by a certain exterior guidance, inasmuch as by sight and the senses, which are rooted in the head, man is guided in his exterior acts.

Christ the Head influences by #1, whereas the "great men, heads of the people" (Amos 6:1), such as bishops, influence by #2.

Christ's headship differs from human headship because

1. Christ is the Head of all who pertain to the Church in every place and time and state; but all other men are called heads with reference to certain special places, as bishops of their Churches. Or with reference to a determined time as the Pope is the head of the whole Church, viz. during the time of his Pontificate, and with reference to a determined state, inasmuch as they are in the state of wayfarers.

2. Christ is the Head of the Church by His own power and authority; while others are called heads, as taking Christ's place, according to 2 Cor. 2:10, "For what I have pardoned, if I have pardoned anything, for your sakes I have done it in the person of Christ," and 2 Cor. 5:20, "For Christ therefore we are ambassadors, God, as it were, exhorting by us."

#2 explains why a Pope is called the Vicar of Christ. "Vicar" is related to the word "vicarious," that which "takes or supplies the place of another thing or person"; a Pope represents Christ.