What is the relationship between state and national government?

I know we have federalism, power which is share between state and national government. However, I can't think of any specific examples as to how this system is in place. Help!

One of the earliest examples was in the Supreme Court's Gibbons v. Ogden decision, which ruled in 1824 that Congress's right to regulate commerce under the Commerce Clause could be "exercised to its utmost extent, and acknowledges no limitations, other than those prescribed in the constitution..." The Court did not expand the powers of the national government much over the next century. But in the 1930's, a wave of feeling of social injustice began to sweep the nation as the Depression began. Federal laws concerning labor, civil rights, and civil liberties began to take on a new priority. National laws, and amendments to the Constitution, have taken away many powers from the states, such as the ability to significantly restrict voting rights or the ability to draw political districts at the whim of political party bosses.

Wiki has a pretty good explanation. Here are some other links that may help.

http://en.wikipedia.org/wiki/Federalism_in_the_United_States

http://www.usconstitution.net/consttop_fedr.html