The Holy Spirit
The Trinity is the central mystery of the Catholic Faith. It is the belief that God is one, yet is composed of three, distinct persons. This is a great mystery, as we cannot understand with our human minds how this is possible. God the Father is often clearly understood, and Jesus, the Second Person of the Trinity, is well known, but the Holy Spirit, the third person, is not so well remembered or understood.
In the Bible
This third person of the Blessed Trinity is talked about in the Bible. Jesus speaks of Him in the gospel of John, particularly at the Last Supper Discourse. “And I will pray the Father, and he will give you another Counselor, to be with you for ever.” - John 14:16. “Nevertheless I tell you the truth: it is to your advantage that I go away, for if I do not go away, the Counselor will not come not you; but if I go, I will send him to you.” - John 16:17. Jesus tells His disciples to wait for and be ready for the coming of the Spirit after He leaves them. In the Acts of the Apostles, the Holy Spirit comes to the Apostles at Pentecost. “When the day of Pentecost had come, they were all together in one place. And suddenly from heaven there came a sound like the rush of a violent wind, and it filled the entire house where they were sitting. Divided tongues, as of fire, appeared among them, and a tongue rested on each of them. All of them were filled with the Holy Spirit and began to speak in other languages, as the Spirit gave them ability.” - Acts 2: 1 – 4. The Holy Spirit is explicitly mentioned here as coming to the Apostles and giving them the courage to proclaim the gospel. This is often what He is associated with: courage and grace to live in accordance with God's will. Finally, the Holy Spirit is mentioned in Scripture in conjunction with the sacrament of Baptism. “And Peter said to them, 'Repent, and be baptized every one of you in the name of Jesus Christ for the forgiveness of your sins; and you shall receive the gift of the Holy Spirit.'” - Acts 2:38. Although He never appears in person in the Bible, the Holy Spirit is very much present and those in the Bible are very much aware of His presence and working in their lives.
Early Church Fathers
As a divine person and a co-equal part of God with the Father and the Son, the Holy Spirit is the subject of many writings of the early church fathers.
Since, also, there is but one unbegotten Being, God, even the Father; and one only-begotten Son, God, the Word and man; and one Comforter, the Spirit of truth; and also one preaching, and one faith, and one baptism; and one Church which the holy apostles established from one end of the earth to the other by the blood of Christ, and by their own sweat and toil. - Ignatius of Antioch The Epistle to the Philadelphians Chapter 4
Thus also we who are baptized, having wiped off the sins which obscure the light of the Divine Spirit, have the eye of the spirit free, unimpeded, and full of light, by which alone we contemplate the Divine, the Holy Spirit flowing down to us from above. - Clement of Alexandria The Instructor Chapter 6
Then there is the Paraclete or Comforter, also, which He promises to pray for to the Father, and to send from heaven after He had ascended to the Father. He is called “another Comforter,” indeed; but in what way He is another we have already shown, “He shall receive of mine,” says Christ, just as Christ Himself received of the Father’s. Thus the connection of the Father in the Son, and of the Son in the Paraclete, produces three coherent Persons, who are yet distinct One from Another. - Tertullian Against Praxeas Chapter 25
Wherefore, when our Lord breathed on His disciples, and said, “Receive ye the Holy Ghost,” He certainly wished it to be understood that the Holy Ghost was not only the Spirit of the Father, but of the only begotten Son Himself. For the same Spirit is, indeed, the Spirit of the Father and of the Son, making with them the trinity of Father, Son, and Spirit, not a creature, but the Creator. For neither was that material breath which proceeded from the mouth of His flesh the very substance and nature of the Holy Spirit, but rather the intimation, as I said, that the Holy Spirit was common to the Father and to the Son; for they have not each a separate Spirit, but both one and the same. - Augustine City of God 13.24
Through the Holy Spirit comes our restoration to paradise, our ascension into the kingdom of heaven, our return to the adoption of sons, our liberty to call God our Father, our being made partakers of the grace of Christ, our being called children of light, our sharing in eternal glory, and, in a word, our being brought into a state of all “fulness of blessing,” both in this world and in the world to come, of all the good gifts that are in store for us, by promise hereof, through faith, beholding the reflection of their grace as though they were already present, we await the full enjoyment. - Basil the Great On the Holy Spirit 15.36
These authors speak both to the distinctness of the Holy Spirit and to His ministry in the world. He is indeed a separate person of the Blessed Trinity, and yet is also fully God. Their writings give great witness to the belief and understanding of the early church on the Holy Spirit.