The doctrine of eternal procession of the Holy Spirit is an attempt to explain how the Holy Spirit relates to the other members of the Trinity. The concept was introduced in the Nicene Creed as revised at the Council of Constantinople to affirm the deity of the Holy Spirit: “We believe . . . in the Holy Ghost, the Lord and Giver of life, who proceeds from the Father, who with the Father and the Son together is worshiped and glorified, who spoke by the prophets.”
According to the Nicene Creed, the Son is begotten (not made) of the Father, and theologians understand this as an eternal begetting. When a person makes something, they make something other than themselves—a sculptor makes a statue. But if that sculptor begets something, it will be another person of the same kind as himself. This is something of the logic behind the creed. The Son is the eternal offspring of the Father. Offspring indicates that He is of the same kind of being as the Father, not a creation, and the fact that He was “begotten” from all eternity means that He is co-eternal with the Father. Admittedly, this is somewhat difficult to understand, but often the sun is given as an illustration. Just as the rays of light stream from the sun, so the Son streams from the Father eternally. Just as the sun and the rays of light co-exist, one is not before the other, so the Father and Son co-exist eternally. This was the logic behind the Nicene Creed.
In the years after Nicaea, the deity of the Holy Spirit began to be questioned. In order to address this, the Nicene Creed was revised at the Council of Constantinople to affirm that the Holy Spirit proceeds from the Father. As the begetting of the Son is eternal, so the procession of the Holy Spirit is eternal. Once again, as rays of light proceed from the sun, so the Holy Spirit proceeds from the Father and, since this procession is from all eternity, the Father is not temporally prior to the Spirit—both are co-eternal.
The eternal begetting of the Son and the eternal procession of the Holy Spirit were the concepts introduced at Nicaea and Constantinople to affirm the deity of the Son and Spirit and try to explain the relationship between the members of the Trinity in a way that would account for the biblical language. This language seems to give priority to the Father as Father yet still affirms the full deity of the Son and Holy Spirit. They are submissive to the Father but in no way inferior to Him.
Wayne Grudem in Systematic Theology: An Introduction to Biblical Doctrine points out that one problem with these kinds of formulations is that they attempt to explain the eternal relationships within the Trinity based on the biblical information that addresses the relationships in time (see p. 246). In John 15:26 Jesus tells the disciples that He will send the Holy Spirit who proceeds from the Father. Is Jesus really explaining the eternal relationship of the Father and Spirit here? Or is He simply telling the disciples that the Spirit will come to them from the Father?
It is possible that Scripture uses the terms Father, Son, and Holy Spirit to explain the way the members of the Trinity relate to us in space-time. In the way the Trinity has chosen to interact with human beings, one Person occupies the Father role, the second occupies the Son role because He was the one who submitted Himself to the Father and was born as a human being, and one occupies the Spirit role because He is the one who empowers believers to live in a way that pleases God. C. S. Lewis envisions this as God the Father who is above us, God the Son who is beside us, and God the Spirit who is in us. If this view is correct, the Holy Spirit proceeded from the Father when He was sent into the world at Pentecost, but that procession was not necessarily ”eternal.” That is, in eternity past, the Spirit existed with the Father and the Son, but He did not "proceed"—He was not "sent"—by the Father. In Genesis 1:2, "the Spirit of God was hovering over the waters"—He was involved in creation, but that says nothing about His “eternal procession.”