What is a state and why are they important?

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).