Fr. Hardon, S.J., gives a good definition of "divine procession:"
The origin of a divine person from another through the communication of the numerically one divine essence. There are two internal processions in the Trinity: the begetting of the Son from the Father, and the procession of the Holy Spirit from the Father and the Son. It is divine persons, not the divine nature, who are the subjects of the internal divine procession. The second divine person proceeds from the intellect of the first divine person by generation, and therefore is related to him as Son to a Father. The third divine person proceeds from the will or mutual love of the Father and the Son as from a single principle through spiration.
“Who Proceedeth from the Father and the Son”
With regard to the words immediately succeeding: who proceedeth from the Father and the Son, the faithful are to be taught that the Holy Ghost proceeds by an eternal procession from the Father and the Son, as from one principle. This truth is proposed for our belief by the Creed of the Church, from which no Christian may depart, and is confirmed by the authority of the Sacred Scriptures and of Councils.
Christ the Lord, speaking of the Holy Ghost, says: He shall glorify me, because he shall receive of mine. (John 16:14). We also find that the Holy Ghost is sometimes called in Scripture the Spirit of Christ, sometimes, the Spirit of the Father; that He is one time said to be sent by the Father, another time, by the Son—all of which clearly signifies that He proceeds alike from the Father and the Son. He, says St. Paul, who has not the Spirit of Christ belongs not to him. (Rom. 8:9). In his Epistle to the Galatians he also calls the Holy Ghost the Spirit of Christ: God hath sent the Spirit of his Son into your hearts, crying: Abba, Father. (Gal. 4:6). In the Gospel of St. Matthew, He is called the Spirit of the Father: It is not you that speak, but the Spirit of your Father that speaketh in you. (Matt. 10:20).
Our Lord said, at His Last Supper: When the Paraclete cometh, whom I will send you, the Spirit of truth, who proceedeth from the Father, he shall give testimony of me. (John 15:26). On another occasion, that the Holy Ghost will be sent by the Father, He declares in these words: Whom the Father will send in my name. (John 14:26). Understanding these words to denote the procession of the Holy Ghost, we come to the inevitable conclusion that He proceeds from both Father and Son.
The above are the truths that should be taught with regard to the Person of the Holy Ghost.
The procession of the Holy Spirit
The procession of the Holy Spirit is one of the four relations in the Holy Trinity (cf. St. Thomas Aquinas's Summa Theologica I q. 28 a. 4).
paternity (the relation of the Father to the Son)
filiation (the relation of the Son to the Father)
active spiration of the Holy Spirit (the relation of the Father and the
Son in respect to the Holy Spirit)
passive spiration of the Holy Spirit (the relation of the Holy Spirit in
respect to the Father and the Son)
There are two processions in the Trinity:
- Generation (the Son proceeding from the Father)
- Procession (the Holy Spirit proceeding from the Father and from the Son)
Relations are depicted in green:
(original image source)
St. Thomas Aquinas describes the procession of the Holy Spirit from the Father and from the Son in his Compendium Theologiæ cap. 49:
We should recall that the act of understanding proceeds from the intellectual power of the mind. When the intellect actually understands, the object it understands is in it. The presence of the object known in the knower results from the intellectual power of the mind, and is its word, as we said above. Likewise, what is loved is in the lover, when it is actually loved. The fact that an object is actually loved, results from the lover’s power to love and from the lovable good as actually known. Accordingly the presence of the beloved object in the lover is brought about by two factors: the appetitive principle and the intelligible object as apprehended, that is, the word conceived about the lovable object. Therefore, since the Word in God who knows and loves Himself is the Son, and since He to whom the Word belongs is the Father of the Word, as is clear from our exposition, the necessary consequence is that the Holy Spirit, who pertains to the love whereby God is in Himself as beloved in lover, proceeds from the Father and the Son. And so we say in the Creed: “Who proceeds from the Father and the Son.”
For more information, see Fr. Gilles Emery, O.P.'s works on the Trinity, e.g., The Trinitarian Theology of St. Thomas Aquinas ch. 5 "Relations" (p. 78 f.); cf. this answer.