Obviously there’s no analogy within our frame of reference that perfectly corresponds to all aspects of the biblical testimony that God is one and yet exists eternally in three persons. So trying to unpack this concept can be like trying to nail down Jell-O! One could say: One in essence – three in person – a personhood that is both diverse and equal. But of course that’s an explanation that needs an explanation. So to flesh that out we could say: The Trinity – or Tri-unity of God – consists of three ideas: unity – diversity – and equality. This unity means that God is one in characteristics (attributes). This diversity means that God is three in persons. And this equality means that those three persons equally possess the same attributes.
It’s critical that we get this right. It’s easy to emphasize unity at the expense of diversity, (one being who appears in three forms – which is modalism). It’s easy to emphasize unity at the expense of equality, (in essence and status the Son is inferior to the Father and the Spirit is inferior to the Father and the Son – which is subordinationism). And it’s easy to emphasize plurality without singularity, (belief in three separate gods – which is tritheism).
Bottom Line: We don’t have to reinvent the wheel when it comes to this stuff. Many of these issues have already been thoroughly dealt with – which is why an awareness of past orthodoxy can help us avoid present heresy.