The Supremacy Clause (Article VI, Clause 2) of the United States Constitution declares that federal laws are the "supreme Law of the Land." Hence, state court laws are inferior so long as the federal law is valid (constitutional).
Given the split jurisdiction between federal and state sovereignty set up by the Constitution, one major point of contention is the 'in pursuance thereof' phrase in the Supremacy Clause. This means that it is not always possible to make a federal law in certain areas - that is, certain areas of law are reserved solely for the states to regulate, and any federal law in that area is unconstitutional. Defining this split of exclusive jurisdiction is the job of the Judicial Branch via interpretation of the Constitution, and has changed over time. In any case, the Constitution itself retains supremacy, so even if federal law is forbidden in certain areas, the Constitution itself always is supreme over any state law.
Example of federal law supremacy
An instance that comes to mind right away is when State laws violate a person's federally enforced rights. Since the passage of the Civil Rights Act of 1964 and the Voting Rights Act of 1965, states are not allowed to have laws that discriminate based on race and certain other factors, even through many of the additional voter eligibility rules can be set by states. Thus, states are forbidden to set voter eligibility based on race, gender, or age (18 is mandated), but, they can set it based on prior criminal record, mental health, etc, up until the federal government passes a law on those subjects, at which point, the federal law takes precedence.
Example of state law supremacy
An instance of this latter case (where there is no federal jurisdiction, but states cannot pass un-Constitutional laws) is the content of Education. Educational curriculum is set by the states - while the Federal government can provided funding if certain other criteria are met, the final say on what curriculum a child is taught is up to the states. However, no state law can violate the Constitution - thus, a state cannot mandate the teaching of a religion in its schools, since that would clearly violate the 1st Amendment to the Constitution.