The short answer to your question is The Holy Spirit is the 3rd person of the Trinity. He is the Person that is with us today both with the Church at Large and with each person individually as said in 1 Cor 16:19 Or do you not know that your body is a temple of the Holy Spirit who is in you, whom you have from God, and that you are not your own? Thus it is He, The Holy Spirit, that dwells among us today.
The Holy Spirit is mentioned throughout the Scripture both Old and New Testament. In fact He is the First Person of the Trinity to be mentioned specifically in Scripture see Genesis 1:2. There is tendency for people to think of the Holy Spirit as the Wind of God as the word Spirit is rooted in the word for wind or breath both in the Hebrew and Greek. However Jesus makes it clear that the Holy Spirit is a Person with a Mission of God in John 16:12-15 12 “I have many more things to say to you, but you cannot bear them now. 13 But when He, the Spirit of truth, comes, He will guide you into all the truth; for He will not speak on His own initiative, but whatever He hears, He will speak; and He will disclose to you what is to come. 14 He will glorify Me, for He will take of Mine and will disclose it to you. 15 All things that the Father has are Mine; therefore I said that He takes of Mine and will disclose it to you. And there are other places in Scripture that specifically identifies the Holy Spirit as a Person not an It.
The ancient Creed of Athanasius I believe is the most explicit description of the Trinity. It is not often referred to as it also explicitly says that if you do not believe and accept the Trinity then you do not believe in God and are condemned. But I put a portion of the Creed below so that you can understand what Christianity believes about the Holy Spirit as I believe there is no better statement about the Trinity, I would remind that the word catholic means universal and is not referring to the Roman Catholic Church but to the Universal Church of Christianity.
Whoever wants to be saved should above all cling to the catholic faith.
Whoever does not guard it whole and inviolable will doubtless perish eternally.
Now this is the catholic faith: We worship one God in trinity and the Trinity in unity, neither confusing the persons nor dividing the divine being.
For the Father is one person, the Son is another, and the Spirit is still another.
But the deity of the Father, Son, and Holy Spirit is one, equal in glory, coeternal in majesty.
What the Father is, the Son is, and so is the Holy Spirit.
Uncreated is the Father; uncreated is the Son; uncreated is the Spirit.
The Father is infinite; the Son is infinite; the Holy Spirit is infinite.
Eternal is the Father; eternal is the Son; eternal is the Spirit:
And yet there are not three eternal beings, but one who is eternal;
as there are not three uncreated and unlimited beings, but one who is uncreated and unlimited.
Almighty is the Father; almighty is the Son; almighty is the Spirit:
And yet there are not three almighty beings, but one who is almighty.
Thus the Father is God; the Son is God; the Holy Spirit is God:
And yet there are not three gods, but one God.
Thus the Father is Lord; the Son is Lord; the Holy Spirit is Lord:
And yet there are not three lords, but one Lord.
As Christian truth compels us to acknowledge each distinct person as God and Lord, so catholic religion forbids us to say that there are three gods or lords.
The Father was neither made nor created nor begotten;
the Son was neither made nor created, but was alone begotten of the Father;
the Spirit was neither made nor created, but is proceeding from the Father and the Son.
Thus there is one Father, not three fathers; one Son, not three sons; one Holy Spirit, not three spirits.
And in this Trinity, no one is before or after, greater or less than the other;
but all three persons are in themselves, coeternal and coequal; and so we must worship the Trinity in unity and the one God in three persons.
Whoever wants to be saved should think thus about the Trinity.