Holy Trinity by Hendrick van Balen (1620sAD)
On this great feast, we might note, in the first place, the great commission of Our Lord, Jesus Christ (Matthew 28:18-20):
"28:18 And Jesus coming, spoke to them, saying: All power is given to me in heaven and in earth. 19 Going therefore, teach all nations: baptizing them in the name of the Father and of the Son and of the Holy Ghost. 20 Teaching them to observe all things whatsoever I have commanded you. And behold I am with you all days, even to the consummation of the world."
Second, the profound collect of the feast:
"O Almighty and everlasting God, who hast enabled Thy servants, in confessing the true Faith, to acknowledge the glory of the eternal Trinity, and in the power of Majesty to adore its Unity: we beseech Thee, that by steadfastness in the same Faith, we may ever be defended against all adversity..."
Next, from the text of the Summa Theologiae of St. Thomas Aquinas (+1274AD), from the Prima Pars, Question 30, Article 2, a reflection on the nature of the Holy Trinity, and the necessity of three persons:
"I answer that, As was explained above, there can be only three persons in God. For it was shown above that the several persons are the several subsisting relations really distinct from each other. But a real distinction between the divine relations can come only from relative opposition. Therefore two opposite relations must needs refer to two persons: and if any relations are not opposite they must needs belong to the same person. Since then paternity and filiation are opposite relations, they belong necessarily to two persons. Therefore the subsisting paternity is the person of the Father; and the subsisting filiation is the person of the Son. The other two relations are not opposed to either of these, but are opposed to each other; therefore these two cannot belong to one person: hence either one of them must belong to both of the aforesaid persons; or one must belong to one person, and the other to the other. Now, procession cannot belong to the Father and the Son, or to either of them; for thus it would follows that the procession of the intellect, which in God is generation, wherefrom paternity and filiation are derived, would issue from the procession of love, whence spiration and procession are derived, if the person generating and the person generated proceeded from the person spirating; and this is against what was laid down above (27, 3 and 4). We must frequently admit that spiration belongs to the person of the Father, and to the person of the Son, forasmuch as it has no relative opposition either to paternity or to filiation; and consequently that procession belongs to the other person who is called the person of the Holy Ghost, who proceeds by way of love, as above explained. Therefore only three persons exist in God, the Father, the Son, and the Holy Ghost."
New Advent: Summa I, Q. 39, A. 2
Of course, while St. Thomas Aquinas, the Angelic Doctor, attempts to explain some aspects of the Holy Trinity, it remains a mystery beyond the created intellect of man! There have been many analogies, of course, to try to express the mystery of the Trinity, such as St. Patrick's clover. One of my favorite analogies for the Trinity is from arithmetic: 1 x 1 x 1 = 1.
You might consult these links for more information on the doctrine and the feast:
Catechism of the Catholic Church: On the Trinity, #232-267
Old Catholic Encyclopedia: Trinity
Old Catholic Encyclopedia: Trinity Sunday
Fisheaters: Trinity Sunday
Finally, a selection, the Credo, from the Trinity Mass of Wolfgang Amadeus Mozart (+1796AD), written for this very Sunday:
Blessed be the Holy Trinity, and undivided Unity! We will give glory to Him, because He hath shown His mercy to us! (Introit of Trinity Sunday)