What is a state and why are they important?
The primary function of a state is to represent the interests of its people. States are comprised of individuals who look to it, "the state," in order to provide for and accurately represent their interests and concerns. It is at this point where the question might need more refinement. "The state" is a term that can describe the institution of government from a theoretical point of view. It can refer to how government should function. For example, "Locke's vision of the state is vastly different than Hobbes' vision of the state." In this statement, "the state" refers to the institution of government. Another definition of "a state" could simply refer to a smaller territory in a larger collection. For example, Louisiana is a state in America, while Bihar is a state of India.
This really depends on what discipline you are talking about.
In international relations, a state is what we usually call a country. It is an entity that is sovereign within its own boundaries and has no higher government that has the right to tell them what to do.
In international relations, the realists say that states are important because they are the only important actors in international affairs. Only states do anything -- individuals, non-governmental organizations, none of these mean anything to the realists.
Within US politics, states are the level of government below the national government. They are important because many of the policies vary by state (things like tax policies and welfare program).
The term state as used in social sciences refers to a group of people occupying a given area under a government. The state can be an independent country or nation, or it can be a a major political subdivision within a countries. For example, countries like U.S.A. and India contain multiple states within them, each having independent state government, that operate within the framework of a central national constitution.
In international politics, the terms nation, country, and states are often used interchangeably.
The word state is also sometimes used to refer to the government machinery of a country or a society.