The Trinity is three separate persons but one God.
We know this through Christ's revelation, but why is it the case?
It may help to realise that the Trinity is the perfect example of Love. Whenever we say 'God is Love' we are proclaiming the reality of what God is.
But if God was one person, how could He be Love? Love is between persons.
There is only one God. But God is Love. Therefore God is more than one person. God is not alone!
Even if we didn't have Christ's explicit revelation of the Trinity we would have to imagine at least a duality of persons 'within' God. And Christ has revealed who these two persons are: God the Father and God the Son.
These divine persons love each other so fully, so completely that their love is also a person, of completely equal status, being and personhood to the Father and the Son. This person is the Holy Spirit.
The closest thing on earth to the Triune reality of God is the Family. The Trinity is Father, Son and Holy Spirit. the Family is Father, Mother, Child.
It's not a perfect analogy, but we can see how the love between two persons in the family becomes a third person, and how a family is incomplete - indeed we can't call it a family - unless it contains (at least) three persons.
Parenthically: this is also why our enemies are always attacking the Family. The Family is the image of the Trinity, so the Devil keeps trying to efface it.
Indeed, we have trouble thinking of the Family as an image of God's love precisely because of this ceaseless assault. Our society is littered with broken images of the Trinity: broken homes, broken marriages, the families killed by the hellish machinations of Planned Parenthood and so on.
It's worth reading Thomas Aquinas on the Trinity and about the nature of personhood and Godhood. Aquinas belonged to a time of intense coherent thought that we moderns can barely understand, so I recommend approaching him through his commentators. Chesterton's biography "Thomas Aquinas" is a good first step.
Hope this was helpful.