I’ve always found this definition from Charles Hodge to be succinct and helpful.

“The Father says I; the Son says I; and Spirit says I. The Father says Thou to the Son, and the Son says Thou to the Father; and in like manner the Father and the Son use the pronouns He and Him in reference to the Spirit. The Father loves the Son; the Son loves the Father; the Spirit testifies of the Son. The Father, Son and Spirit are severally subject and object. They act and are acted upon, or are objects of action. Nothing is added to these facts when it is said that the Father, Son and Spirit are distinct persons; for a person is an intelligent subject who can say I, who can be addressed as Thou, and who can act and be the object of action. The summation of the above facts is expressed in the proposition, The one divine Being subsists in three persons, Father, Son and Holy Spirit. This proposition adds nothing to the facts themselves; for the facts are (1.) That there is one divine Being. (2) The Father, the Son and Spirit are divine. (3.) The Father, Son and Spirit are in the sense just stated, distinct persons. (4.) Attributes being inseparable from substance, the Scriptures, in saying that the Father, Son and Spirit possess the same attributes, say they are one in substance; and, if the same in substance, they are equal in power and glory”

From Charles Hodge, Systematic Theology, I.444

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