Three examples of cooperative federalism at work: 1.) Federal grants-in-aid programs and other block grants. 2.) State aid in conducting, financing, and regulating elections. 3.) State aid in helping handle federal criminals.
Cooperative federalism is a political and constitutional concept that emphasizes the decentralization of power and a not necessarily equal sharing of governmental responsibilities between federal, state and local agencies and constitutions.
Federalism in the United States has evolved quite a bit since it was first implemented in 1787. In that time, two major kinds of federalism have dominated political theory.
The first, dual federalism, holds that the federal government and the state governments are co-equals, each sovereign. In this theory, parts of the Constitution are interpreted very narrowly, such as the 10th Amendment, the Supremacy Clause, the Necessary and Proper Clause, and the Commerce Clause. In this narrow interpretation, the federal government has jurisdiction only if the Constitution clearly grants such. In this case, there is a very large group of powers belonging to the states, and the federal government is limited to only those powers explicitly listed in the Constitution.
The second, cooperative federalism, asserts that the national government is supreme over the states, and the 10th Amendment, the Supremacy Clause, the Necessary and Proper Clause, and the Commerce Clause have entirely different meaning. A good illustration of the wide interpretation of these parts of the Constitution is exemplified by the Necessary and Proper Clause's other common name: the Elastic Clause.
For further reading, go to USConstitution.net.