Nature of Agape (Catholic)

In Cardinal Levada’s glossary, charity (or love) is defined as, "The theological virtue by which we love God above all things for his own sake, and our neighbor as ourselves for the love of God." Paul tells us that love is the greatest of the theological virtues: "So faith, hope, love abide, these three; but the greatest of these is love" (1 Cor. 13:13).

Our Sunday Visitor’s Catholic Encyclopedia entry on "Love" defines it as:

[A]ny strong affection, closeness, or devotion to things or persons. The Greeks distinguished four types of love: storge, philia, eros, and agape. Storge, familial love, is a word for the bond that exists between one who loves and persons, animals, and the things that surround him. It is compatible with quite a bit of taken-for-grantedness or even of hatred at times. Philia pertains to friends, freely chosen because of mutual compatibility and common values. Eros is passion, not only of a sexual nature, but also of an aesthetic or spiritual nature, for what is conceived of as supremely beautiful and desirable. Agapic love is manifested when one person has much to give to another more needy. It is generous self-donation without concern for reward. Such distinctions become especially important in discernments about marriage, because the strength of eros love may blind one to the absence of ther types of love needed to experience a good Christian bond that, with God’s grace, can endure "till death do us part."

My addendum:

Agape ~ Caritas ~ Charity

https://en.wikipedia.org/wiki/Charity_(virtue)

Note:

Taken straight from Catholic Answers

http://www.catholic.com/quickquestions/how-does-the-church-define-love

Does agape contain affection for others whom the agape is directed to?
Short answer: Yes(it can, but it does not seem to be required to).

Discussion
Definition: Agape (Ancient Greek: ἀγάπη) is "love: the highest form of love, charity; the love of God for man and of man for God."

The Catholic teaching on Charity.

The catholic teaching is that agape/charity, is the greatest theological virtue. From the Catechism:

1822 Charity is the theological virtue by which we love God above all things for his own sake, and our neighbor as ourselves for the love of God.
1823 Jesus makes charity the new commandment(John 13:34). By loving his own "to the end," (John 13:1) he makes manifest the Father's love which he receives. By loving one another, the disciples imitate the love of Jesus which they themselves receive. Whence Jesus says: "As the Father has loved me, so have I loved you; abide in my love." and again: "This is my commandment, that you love one another as I have loved you." (John 15:9; 12)

1824 Fruit of the Spirit and fullness of the Law, charity keeps the ommandments of God and his Christ: "Abide in my love. If you keep my commandments, you will abide in my love."
1825 Christ died out of love for us, while we were still "enemies." The Lord asks us to love as he does, even our enemies, to make ourselves the neighbor of those farthest away, and to love children and the poor as Christ himself.

The Apostle Paul has given an incomparable depiction of charity: "charity is patient and kind, charity is not jealous or boastful; it is not arrogant or rude. Charity does not insist on its own way; it is not irritable or resentful; it does not rejoice at wrong, but rejoices in the right. Charity bears all things, believes all things, hopes all things, endures all things." (1 Corinthians 13).

1826 "If I . . . have not charity," says the Apostle, "I am nothing." Whatever my privilege, service, or even virtue, "if I . . . have not charity, I gain nothing." Charity is superior to all the virtues. It is the first of the theological virtues: "So faith, hope, charity abide, these three. But the greatest of these is charity." (1 Corithians, 13:13)

St Basil:

If we turn away from evil out of fear of punishment, we are in the position of slaves. If we pursue the enticement of wages, . . . we resemble mercenaries. Finally if we obey for the sake of the good itself and out of love for him who commands . . . we are in the position of children.

St Augustine:

Love is itself the fulfillment of all our works. There is the goal; that is why we run: we run toward it, and once we reach it, in it we shall find rest.

On an experiential note, in the past six years I have been involved with a lot of lay retreats that place heavy emphasis on the virtue of charity, and the principle of agape as the theme of the retreats. Agape is alive and well, and being emphasized.