How much autonomy do the States Constitutions have over the Federal in U.S?

Question

I am carrying out a research on Federal Constitutions. In Brazil, for instance, although brazilian states(27) having their own Constitution, they are blindly and obliged to follow the Federal one (despite the possibility of amending).

In U.S, how does it work the States Constitutions and their "independence" towards the Federal one?

Reminder: best answer will be rewarded with 10 points!

Regards

Answer 1

State governments have soveriegn powers within their own border but things outside of the state governments such as disputes over water rights or declaring war on a foriegn power is handled by the federal government.  

Answer 2

None. In the U.S. the Federal Govt's laws override State laws. If there is no Federal Law then State Law is what counts. Here in the the U.S. the laws are set by Federal, then State, and last is Local. The higher level always overrides. Although in the U.S. the Constitution can still be amended.