What is the due process clause and how has it been used by the judiciary to apply the Bill of Rights to the actions of the state governments?
There is a due process clause in the Fifth Amendment and another in the 14th. The clause in the 14th Amendment is the one that is relevant to what you are talking about.
Before the Civil War (and really up until around WWI) the protections of the Bill of Rights only "worked" against actions of the federal government. The states could violate the rights given in the Bill of Rights if they wished.
After WWI, the Supreme Court gradually started a process of "selective incorporation" by which they applied most rights to the actions of the states. Here's how that works:
The clause says states can't deprive people of liberty without the due process of law. The Court has decided that most rights from the Bill of Rights are so important that they are part of this "liberty." Therefore, these fundamental rights cannot be taken away without due process of law (basically, a trial in which you are convicted of a crime).
"The Fourteenth Amendment reads, in part, that no state shall 'deprive any person of life, liberty, or property, without due process of law.' This applies to the states and to local governments. The Due Process Clause of the Fifth Amendment applies to the federal government. Most Due Process issues involve state laws." - nationalparalegal.edu
Substantive Due Process: Involves regulation by the states powers.
Procedural Due Process: Involves regulation required by the Constitution.
Please read this guide: Substantive Due Process- Fundamental Rights It contains some great examples in laymens terms to help students fully grasp and understand these clauses.