There is no doctrine more essential to understanding Christian theology than the trinity yet at the same time it is by far the most difficult doctrine to understand and explain. Emil Brunner called it the “theological doctrine which defends the central faith of the Bible and the Church.”
The doctrine was first defined as a way to navigate between two false teachings: tritheism and Sabellianism. Tritheism is the belief that the Father, Son, and the Holy Spirit are three completely different gods. Sabellianism claims that there is only one God and that the Father, Son and Holy Spirit are different ways in which He has revealed Himself at different times. It denies the uniqueness or independence of the various members of the trinity. Although none would claim it today, most dispensationalist theologians have their roots in Sabellianism.
In an effort to guard against these two heresies three theologians, now known as the Cappodacian fathers, used two Greek words: ousia (essence), and hypostasis (independant reality). God is one ousia but three hypostasis. God is one in nature, essence, and will but each member of the trinity is an “independent reality” that has their own special properties and activities.
Every analogy limps and I am not a big fan of any analogies used for the trinity but two that are most commonly used are the egg and water. There is one egg but it is composed of a shell, the white, and the yolk. Water is always in “essence” H2O but it can be found in any of the three “independent realities” of ice, liquid, or steam.