What “ontological” means
Ontological is a long word for a simple question: what is something in its deepest being? So when people speak of the ontological Trinity, they mean the Trinity as God eternally is, before the world was made and apart from what God does in history.
This is different from asking how the Father, Son, and Holy Spirit work in the world. The Father sends the Son, the Son becomes human and saves, and the Spirit gives life and strengthens believers. Those actions matter deeply, but the ontological Trinity asks who these three are in relation to one another.
One God, three persons
The Bible holds together two truths. God is one, and yet the Father, Son, and Holy Spirit are each spoken of personally and in ways that belong to God. Jesus’ baptism shows the Son in the water, the Spirit descending, and the Father speaking from heaven at the same moment (Matthew 3:16-17). They are not simply three names for one person acting in different ways.
Jesus also gives one baptismal name that includes the Father, the Son, and the Holy Ghost (Matthew 28:19). Paul closes his letter to Corinth by placing the grace of Christ, the love of God, and the fellowship of the Holy Ghost together in blessing (2 Corinthians 13:14). These passages do not lay out a technical definition, but they help Christians recognise the pattern.
Go ye therefore, and teach all nations, baptizing them in the name of the Father, and of the Son, and of the Holy Ghost:
The Son has eternal glory
The ontological Trinity means the Son is not a created being who later became divine. John introduces the Word as existing in the beginning, being with God, and being God, and says that all created things came through him (John 1:1-3). John identifies this Word with Jesus Christ a little later (John 1:14).
On the night before his crucifixion, Jesus speaks of the glory he shared with the Father before the world existed (John 17:5). Paul likewise describes Christ as existing in the form of God before taking the form of a servant (Philippians 2:6-7). Christians therefore confess that the Son is eternally divine, not less than the Father.
In the beginning was the Word, and the Word was with God, and the Word was God.
The Spirit is not an impersonal force
The Holy Spirit is also more than God’s power or influence. Jesus speaks of the Spirit as one who teaches and brings his words to the disciples’ remembrance (John 14:26). In the account of Ananias, Peter says that lying to the Holy Ghost is lying to God (Acts 5:3-4).
So the usual Christian shorthand is this: one God, three persons, equal in divine being and eternal in relationship. The word Trinity itself is not found in the Bible, but it is a useful label for gathering what Scripture says. It does not solve every mystery. It helps guard the truth that Christians worship the Father, through the Son, in the Holy Spirit, without dividing God into three gods.