One of the first things to understand is that basically all analogies are going to be problematic. It's why trying to say "The Trinity is like..." usually ends up just causing more confusion than bringing clarity.
God as Trinity isn't like anything with which we have in our experience. There's simply nothing like God.
So when we say that there are three Divine Persons, each fully and equally God--the one and same God--it describes something which, on some level, we simply can't relate to on the basis of our own experiences outside of God's own self-revelation of Himself.
Two things are being asserted as fundamentally true:
1. There is only one God.
2. There are three Persons or Hypostases.
That word hypostasis (plural hypostases) is a complicated one. It's a Greek word that can be translated as "subsistence". But what it is really trying to capture is the idea of meaningful existence, something discrete and real. In regard to the Trinity it means that the Father is really the Father, that's Who He is; distinct from the Son and the Holy Spirit. "Father" isn't just a face, or a mask God wears, instead there really is this One who is Father, and He is Father because He is the Father of the Son. It is therefore relational, He is the Father of the Son, that is very real, eternally real. In the same way that the Son, distinct from the Father and the Holy Spirit, really exists as the Son of the Father, the only-begotten of the Father, begotten of the Father from all eternity.
So that there is one God, the Father. And the Father eternally begets the Son, and the Son as the eternally-begotten of the Father is God, just as the Father is God. So that the Father and the Son are the same God.
The Son is God
because the Father is God. Thus we speak of the Son's eternal generation from the Father, as God of God, and therefore He is homoousios (of-Same-Being) with the Father. The Father's Being, His Essence, His Eternal
IS-ness as the one and only God is also the Being, Essence, and Eternal
IS-ness of the Son. Thus the Son is God, the one and only God, even as the Father is God, the one and only God.
Likewise, the Holy Spirit eternally proceeds from the Father [and the Son], and is therefore God, in just the same way the Father and the Son are God. The one and only God.
So that there is the Father, unbegotten and unproceeding.
The Son, only-begotten of the Father.
The Holy Spirit, proceeding from the Father [and the Son].
So that the Father, the Son, and the Holy Spirit, three Divine Persons and Hypostases, of one undivided Being and Essence. One God, Three Persons.
Helpful resources:
The Nicene Creed
The Nicene Creed
The Athanasian Creed aka The Quicumque Vult
Quicumque
Perhaps somewhat more obscure, but a local synod held in Toledo in the 7th century produced a rather fantastic statement of Trinitarian faith:
COUNCIL OF TOLEDO XI 675 – Creed of Faith
-CryptoLutheran