("All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside. No State shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States; nor shall any State deprive any person of life, liberty, or property, without due process of law; nor deny to any person within its jurisdiction the equal protection of the laws.")
The individual states are still allowed to administer themselves in uninteresting and uncontroversial areas, but they have no right to legalize dueling, or ban guns, or write their own marriage laws, or set up a state religion, or reorganize themselves as anti-clerical theocracies -- matters where a federal government would presumably let them do as they pleased. (See what I mean about parts of the federation coming to hate each other enough to go to war?)
it's in the constitution, the 10th amendment: "The powers not delegated to the United States by the Constitution, nor prohibited by it to the States, are reserved to the States respectively, or to the people."