In general legal terms, due process is one of the fundamental rights that Americans have. It is protected by the 5th and 14th Amendments to the Constitution, which say that governments may not deprive people of their life, liberty, or property without the due process of law. This means that the government may not take anything away from you without going through the proper procedure of putting you on trial and convicting you of a crime (or the proper procedure of passing a law to impose a tax). In short, proper procedures must be followed to ensure that you are not being punished arbitrarily or unreasonably.
In the workplace, due process means much the same thing, except that here it is the employer being enjoined against abusing workers rather than the government being enjoined against abusing citizens. Due process in the workplace means that your employer cannot punish you without going through properly set-up procedures and without showing in those procedures that there is a valid reason to do so.
For example, in firms where workers have due process rights, the firm has to have written rules about what can and cannot be done. You cannot be punished for doing something that is not prohibited because that would be arbitrary. If you are accused of having done something wrong, there are procedures that the firm must follow to inform you of what you have done. You must have the right to appeal and to dispute the allegations. These rights ensure that the firm must have proof of what you have done in order to punish you.
Thus, due process in the workplace is the right of the employee to be protected against arbitrary and unreasonable punishments.