This seems related to Latin language only per accidens.
I'm not a theologian, but I think I know the basics from a Catholic POV.
I'll try to answer in the order you put the questions:
- What does this mean exactly?
Translations to modern languages vary, but some of them say Christ was conceived by the power of the Holy Spirit. Conception needs the participation of two, undoubtedly. In the case of Jesus, God and man, the preposition de has a double meaning, in words of Thomas Aquinas:
Conception is not attributed only to the body of Christ, but also to Christ himself by reason of his body. Now in the Holy Spirit we can see a twofold relation to Christ. In relation to the son of God, who is said to be conceived, he stands as consubstantial; but in relation to his body he is an efficient cause. The preposition 'of' [de] conveys both relations: as when we say a man is 'of' his father. So we can reasonably say that Christ is conceived of the Holy Spirit, in the sense that the causality of the Holy Spirit is in respect to the body assumed, the consubstantiality in respect of the Person assuming it. (Summa Theologiae, q. 32, art. 2)
Catholic teaching says that Christ is perfectus Deus, perfectus homo, (perfect/complete --truly-- God and man.) There is only one God in three Divine Persons (distinct, but consubstantial, i.e., sharing the same substance): the Father, the Son and the Holy Spirit. In the instant of incarnation, a human body (and its nature) was formed by the power of the Holy Spirit and assumed by the person of God the Son, who is still one person, but holds both natures (cf. Cathechism of the Catholic Church, 464 ss.)
- If Jesus "was conceived by the Holy Ghost," does that mean He was not conceived by The Virgin Mary?
No, it does not. The Gospel of St. Luke (at least since the Vulgate, Lc 1,31) uses the active voice of concipio (if someone can help me with the Greek, it will be appreciated) with the Virgin Mary as the subject: ecce concipies in utero et paries filium. St. Thomas cites Augustine and adds:
As Augustine says (Enchiridion xl), Christ is said to be conceived or born of the Holy Ghost in one sense; of the Virgin Mary in another---of the Virgin Mary materially; of the Holy Ghost efficiently. Therefore there was no mingling here (ibid.)
- Who, exactly, conceived Jesus?
- How should I understand conceptus de Spiritu Sancto in this context?
I think most of these have already been answered in the previous two questions. I only want to add a comment that may be helpful in putting all this in modern context.
The scriptures weren't intended as a scientific document, but as something helpful for salvation to be understood by peoples of all eras. Hence the language must be understandable to everyone. Today we have both science and exegesis, in antiquity they had none. That is why the fact that explanations are given through ancient eyes and language is not to be seen as contradictory to modern knowledge.
In antiquity, it was believed that in the act of conception, the man put the seed, while the woman's womb and blood were the ground and nutrients needed for that seed to grow. I doubt they didn't realize that children inherit treats from both parents, but they didn't know genetics either, so it's reasonable to think this was not a problem for them. Both through the eyes of antiquity and modern biology a child is not produced by the female alone. The missing part was put there de Spiritu Sancto.