Gloria Patri

Glória Patri et Fílio et Spirítui Sancto.

Glory to the Father, and to the Son, and to the Holy Spirit.

The doxology employs datives for the names of the persons of the Trinity, meaning “to / from.” 

Gloria means glory (nominative case).

Patri is from Pater (as in Our Father, Pater Noster). Here its form is dative (meaning to the Father).

Filio is the dative of Filius. Remember David crying out after the death of Absalom (2 Sam 19:1), “Fili mi Absalom, fili mi, fili mi Absalom! Quis mihi tribuat, ut ego moriar pro te? Absalom fili mi, fili mi!” / “My son Absalom, my son, my son Absalom! Who will grant me, that I may die for you? Absalom my son, my son!” Fili = vocative, thus he is calling out to his son. In the doxology it is dative.

Spiritui Sancto, again, is dative = to the Holy Spirit.

How would you say ‘The Son is the glory of the Father?’ = Filius est gloria Patris.

How would you say ‘The Spirit of the Father and of the Son?’ = Spiritus Patris Filiique.