Much of what follows comes from the Westminster Shorter Catechism.
The Father, Son and Holy Spirit are 3 separate Persons (each have their own consciousness, will etc). They partake of one essence (that is what makes each Person to be God).
The essence of God must never be confused with the 3 Persons Who are each God. The essence of God is not a person, otherwise we have 4 separate persons Who are God. That is not what the Bible teaches. The Bible teaches one essence and 3 Persons.
Essence can be applied to anything. It is akin to a definition. Unfortunately, it is almost impossible to define anything since we are so slow of understanding. For example, what is a chair? Does it have 3 legs and a round top? What about 4 legs and a square top? What about steel tubes that curve around...and so on and so on. I'm sure you get the picture.
So moving on from chairs, what is a rock? What is a tree? What is a human? Does a human need two arms? Does a human have to be >3 feet tall? Does a human have to have a certain IQ? You see in all these examples we are trying to understand what set of characteristics make up what we are trying to define. ie what is the essence of a chair (or a rock or a tree or a human). You will see that the essence of anything is extremely hard to define, even for a chair! Yet, the essence has to be complete in order for that thing/person to be really all that thing/person is. If you take away part of the definition (essence) then you have something else. eg take away the FULL definition of a chair (rock, tree etc) and you no longer have a chair, you have something else whatever that may be, but you no longer have a chair.
So apply this to God. What does it mean to be God? Well, in a very small way it means to be a Spirit Who is infinite and eternal in His being, wisdom, power, holiness, justice, goodness and truth. (obviously not a full definition). It is extremely hard to get a full definition of what it means to be God. But scouring the Bible leads us to at least a summary as just given.
Now the 3 Persons, Father, Son and Spirit each have ALL the characteristics just mentioned (infinite, eternal etc). And ONLY these 3 Persons have those characteristics. Nothing else at all has these characteristics, otherwise that thing would also be God. These characteristics comprise the essence of what it means to be God.
So if a person has ALL of these characteristics, then that person must be God.... Just as if any object has ALL the characteristics of a chair, then we say that thing MUST be a chair. It cannot be a tree, since it does not have all the characteristics of what it means to be a tree.
That is also why I can say without reservation that you and I are not God. We simply do not have any of those characteristics and we certainly don't have all of them. But the Father, Son and Spirit each have ALL of those characteristics summarised above. Thus the Father is God, the Son is God, the Spirit is God. They are equal in power and glory.
For Further Reading
Calvin in the 1500's taught what the Bible has to say on the subject (see his Institutes, Book 1 Ch13 and you can go to any good Systematic Theology to find the same answer (eg Charles Hodge, Louis Berkhoff)...