A Brief Explanation of the Trinity - What does this doctrine teach?
This doctrine teaches that God exists in three persons who share the same essence or being. What this means is that God exists in the distinct and co-equal persons of the Father, the Son, and the Holy Spirit, but they share a fully divine essence or being, such as uncreatedness, eternality, simplicity (non-composite or indivisible), immutability (unchangeableness), omniscience (all wise and knowing), omnipotence (all powerful), goodness, mercy, holiness, will and freedom, and so on.
Thus, in God, the attributes of his essence are fully shared by three persons, making each person fully God.
Yet, the three persons are distinct. The Father is neither the Son nor the Holy Spirit; the Son is neither the Father nor the Holy Spirit; and the Holy Spirit is neither the Father nor the Son.
But, to repeat, the three persons share the same essence. That is, the Father and Son and Holy Spirit duplicate the same attributes. The Father, the Son, and the Holy Spirit are equally wise and merciful and holy; one is not more wise or merciful or holy than the other. Augustine says: "These three have the same eternal nature, the same unchangeableness, the same majesty, the same power" (On Christian Teaching I.12).
This site has a list of the divine attributes that all three persons of the Trinity share in perfect unity; click on any of the pages. This page at the same site also references verses in the Bible concerning the divine attributes.