Tertullian did not have the benefit of the Johannine comma (1 John 5:7), so he appears to have taken a bold step forward in talking about the Trinity. Roger E. Olson, Christopher Alan Hall (The Trinity, page 31) say that Tertullian stumbles in his attempts to explain trinitarian relationships, but despite occasional missteps is a reliable guide and avoids the twin hazards of tritheism and modalism.
He believed that God, who made the world out of nothing through his Son, the Word, but Jesus was created, not eternally pre-existent - as we see in Against Praxeas chapter 6, where Jesus had been in the Father's mind as part of the eternal plan (my emphasis):
He first put forth the Word Himself, having within Him His own inseparable Reason and Wisdom, in order that all things might be made through Him through whom they had been planned and disposed, yea, and already made, so far forth as (they were) in the mind and intelligence of God.
In speaking of the Trinity, he said that the Son is distinct from the Father, and the Spirit from both the Father and the Son (Against Praxeam, 25):
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. These Three are one essence, not one Person, as it is said, I and my Father are One, in respect of unity of substance not singularity of number.
He further distinguished the Father, the Son and the Paraclete (Holy Spirit) in Against Praxeam, 13:
For we, who by the grace of God possess an insight into both the times and the occasions of the Sacred Writings, especially we who are followers of the Paraclete, not of human teachers, do indeed definitively declare that Two Beings are God, the Father and the Son, and, with the addition of the Holy Spirit, even Three, according to the principle of the divine economy, which introduces number, in order that the Father may not, as you perversely infer, be Himself believed to have been born and to have suffered, which it is not lawful to believe, forasmuch as it has not been so handed down.
Then, Tertullian saw Jesus as subordinate to the Father, which in most respects is very much contrary to modern theology, as briefly described below.
Some centuries after the Nicene Creed was formulated, the Catholic Church introduced the filioque clause to the Nicene Creed, stating that the Holy Spirit comes from the Father and from the Son, a subtle novelty that Tertullian certainly never taught. This clause was rejected by the Orthodox east.
Eusebius of Caesarea still saw the Nicene Creed as a full expression of Subordinationist theology, but this view was soon rejected. Roman Catholicism is firmly non-Subordinationist, in the Arian or Semi-Arian sense, but does accept a relational subordinationism of Son to the Father. Different views are held amongst Protestants, but most do not differ significantly from the Catholic position.
This site discusses the Catholic doctrine that the second person of the Trinity was the Son of God from all eternity and also criticises the view of some evangelicals that the second person of the Trinity only became the Son of God at his incarnation. The Catholic position differs from that of Tertullian but we see that other, minority views do still exist.
According to the Orthodox Church, the Son or Word is derived from the Father who alone is without cause or origin. This is not a subordination in time, since the Son is co-eternal with the Father or even in terms of the co-equal uncreated nature shared by the Father and Son.