This amendment is among the most important parts of the Constitution. Up to this time, the Constitution didn't define "citizen", so section 1 defines citizenship, extending it to African Americans. Due process of law is the assurance that a law must be applied in a fair manner. Guaranteeing due process and equal protection under the law prevented states from denying African Americans their civil rights. Over the years, the Supreme Court has consistently interpreted the due process clause to apply the guarantees of the Bill of Rights to state governments. They have also extended the meaning of due process to include protection from unreasonable search and seizure, the right to an attorney, and protection from cruel and unusual punishment.
The due process clause says that no person shall be deprived of their life, liberty, or property without the due process of the law. What this means is that a person cannot have any of these things taken away without being convicted of a crime.
Beginning in the early 1900s, the courts have used this clause to say that the Bill of Rights applies to state governments. They have said that the rights from the Bill of Rights are so important that they are part of the "liberty" mentioned in the clause. Because they are part of the "liberty" they cannot be taken away by state governments.
The clause reads that no state can "deprive any person of life, liberty, or property, without due process of law."
Various guarantees in the Bill of Rights create zones known as penumbras, that establish a right to privacy. In the case discussed below, the right to privacy in marital relations was conflicted.
Griswold v. Connecticut (1965). The court found that Connecticut had violated the Fourth Amendment right to privacy along with other portions of the Bill of Rights and the statute was struck down.
Resources on the clause and other cases for reference:
Griswold v Connecticut: Overview of the case involving the Fourteenth Amendment and the violation by a state government of the Bill of Rights.
National Paralegal: Explanation of substantive due process and fundamental rights.
FindLaw: U.S. Constitution, Fourteenth Amendment text and annotations.