The Due Process Clause prohibits state and local governments from depriving persons of life, liberty, or property without a fair procedure. The Supreme Court has ruled that this clause makes most of the Bill of Rights as applicable to the states as it is to the federal government, as well as to recognize substantive and procedural requirements that state laws must satisfy.
It's not straightforward from reading the amendment, so you have to look at Supreme Court rulings to see how it has been interpreted.