What is the sharing of power between the local,state, and national government?
This is a very complicated question.
In general, the Constitution spells out which powers are given to the states and which are given to the national government. States then get to decide what powers their local governments have.
The national government gets most of its power over the states from its power to regulate interstate commerce. This is, for example, why the national government can ban discrimination -- it can ban it in any establishment that is part of interstate commerce (and nearly everything is).
The states have pretty much exclusive power over things like voting rules, taxes, and criminal laws. So, for example, Congress could not pass a law banning the death penalty in the states. It could end the death penalty for federal crimes, but not for state crimes.
However, the national government can exert power over the states by threatening to withhold money if the states don't do what it wants. This is how the No Child Left Behind law works. States generally get to do what they want in education, but the national government says "if you don't do it our way, we won't give you any of this money that we give out to the states."
I'm assuming you mean how is the power shared between the three levels of government at the local, state and national levels - also known as "federalism".
Government at each of these three levels has the power to make laws, hold elections, raise taxes and a number of other things. The Constitution was framed to allow this mainly because it preserves the rights of states and communities to determine their own standards and practices, while giving the federal government the authority to take care of the nation's business.
It's important to note that the Constitution is the Supreme Law of the Land, and all levels of government must follow what it says. The Federal government is next strongest, and all state and local laws must follow both federal law and the Constitution. All local laws have to follow state laws, and so on.
One of the premises of the American Constitution is the idea of federalism. This means that power is to be brokered and shared between local, state, and federal forms of government. The framers of the Constitution understood that the act of governing takes many forms on many levels. In the hopes of preventing the abuses of a strong central government, as in what caused the problems with England and King George, the idea of sharing power between multiple agencies was designed to preclude an overall abuse and misuse of power. There are specific duties that all forms of government share, such as the power to tax and make laws, but the notion of creating institutional power on multiple levels was meant to empower all levels of being able to govern without the premise of abuse.