The Trinity is a description of three unified, yet distinct forms: God the Father, God the Son, God the Holy Spirit. That is to say that, God the Father IS God. God the Son IS God. God the Holy Spirit IS God. They are unified in One, yet they are also distinctly their own 'personalities'. Simply, the Trinity is only graspable--only comprehendable by Faith. I do not believe that the Trinity is a mystery that is designed for us to be able to FULLY grasp here in earth. We need to be sure that we do not fall into various different heresies in describing the Trinity however.
Modalism is a common heresy employed by people trying to describe the Trinity. They say that (in Modalism), there is one God who manifests in different 'modes' or forms. Under this heresy, there is not the concept of union. It flies in the very face of the description of the trinity. By saying there is only one God manifesting, then there is no true distinction while concurrently being God. If one God is simply manifesting in different forms, there is no longer the ability to be all three parts at the same time. There is also not the ability to be FULLY God, if this God is manifesting in different forms...to manifest in different forms implies at its core that this God is not the embodiment of the Trinity...otherwise He would concurrently BE all three parts of the Trinity and would have no need for manifesting.
Describing the Trinity without falling into heresies is not as simple as it seems...it is something that is 'above our paygrade' as humans, and we will likely never have a full, proper understanding of the Trinity this side of Glory.