Throughout U.S. history, there has been a lot of conflict over how much power the federal government should have, and how much states should have. Even today this is a hotly debated subject. At this point the federal government has significantly more power.
From about 1820 to 1860, the states and federal government fought over one specific issue: slavery. Several southern states thought they had the right to decide whether slavery was legal in their states, and that the federal government should have no say in that decision. In fact, it was this conflict that led to the Civil War.
The Fourteenth Amendment, which was passed after the Civil War, banned states from depriving people of their life, liberty, and property without "due process of law." People were also granted "equal protection of the laws." These two clauses gave the federal government to exert the Bill of Rights over state governments. It also ultimately ended segregation of African Americans.