What is the Trinity?

The doctrine of the Trinity has traditionally been formulated with the language "three persons, one essence." It's vague because the Bible is vague on this point. The formulation is meant to convey the following ideas:

1. There is only one God.
2. The Father is God, the Son is God, and the Holy Spirit is God.
3. The Father, the Son, and the Holy Spirit are all distinct persons from one another (as different people are distinct, not as someone with a split personality).

When I think of the word "entity," I tend to think of "being." If that's what you mean, then I would say the that God is one "being" ("being" is closer in my mind to "essence" than it is to "person"). Really, though, it's hard to describe the Trinity with conventional words because we generally come up with words only for those things with which we are familiar, and trinitarian existence is definitely beyond the realm of our familiarity. There is nothing in our normal earthly experience that exists in trinity, and that makes it very difficult to conceptualize.

Imagine if you and two other people shared only one life between you. Each of you would be fully alive, and that life would belong entirely to each of you, but not solely to each of you. None of you would have only 1/3 of a life; each would have a whole life. And yet, there would be only one life and three people.

Or maybe it would help to think of the title to a house. There is a type of ownership in which every owner of the house individually owns the whole house, not just part of it. It might seem from a legal standpoint that more than one house is owned, but in fact there is only one house. Such ownership would be similar to the Trinity (in which there are three persons who are all one God).