The Constitution sets out various powers that are given to the national government and others that are reserved for the states. In addition, it stipulates that the laws passed by the national Congress (as well as the Constitution) are the supreme law of the land and will override state laws.
There are a number of powers that are either specifically reserved for the federal government or kept for the states. For example, the right to conduct foreign policy is reserved for the national government. This can be seen in such places as Article I, Section 10 which says that no state may enter into a treaty with a foreign county. By contrast, it is the states that get to set the rules for who gets to vote in their elections. Finally, the 10th Amendment specifies that all powers and rights not specifically given to the federal government or denied to the states are reserved to the states.
These are a few of the ways in which the Constitution determines the respective powers of the state and national governments.