My layman's 0.02 cents:
God indeed exists as three Divine Persons with one Divine "essence" (Greek "ousia").
Contrary to our human empiric idea that one person ("hypostasis") is necessarily a separate being, distinct from the being of any other person, - God is one Being in three distinct persons ("hypostases"). This means that while John, Jack, and Jill are three persons and three separate human beings, God is three Persons and yet one Divine Being. This also means that Father, Son, and Holy Spirit are not three separate gods, but One God.
The three Persons of one Divine Being eternally "dwell" within each other (if I am not mistaken, there is a Greek term for this, "perikhoresis" - right?). It means that not only their substance or "essence" (ousia) is the same, - their will, their actions in the world and in the human history are also principally the same. Some theologians compared this with a dance, performed by three people who tightly embrace each other and move in absolute, perfect unity.
On the other hand, there are fundamental, eternal distinctions between the three Divine Persons in that Father is the source, the "fountain" of the Two other. His Son, Logos, Christ is being eternally, extemporanously "born" or "begotten" from Him, and the Holy Spirit also eternally, extemporaneously "proceeds" from Him. We do not have any empiric analogies from this "birth" and for this "proceeding."
Now, I wrote this from memory and did not cheat looking at other more learned people's answers. Am I anywhere near?