It is the promise that the government cannot take your freedom without following the rules. And the phrase is older than Parliament, older than England's common law itself.
Due process means the government must follow fair, regular procedures before it deprives any person of life, liberty, or property. Notice, a hearing, an impartial judge: the basic machinery of a fair day in court.
Its ancestor is a single line in Magna Carta, 1215: no free man shall be seized or imprisoned except by the lawful judgment of his peers or by the law of the land. By the law of the land slowly became, over centuries, due process of law.
It appears twice in the Constitution. The Fifth Amendment applies it to the federal government, and the Fourteenth, after the Civil War, applies it to the states. That repetition is deliberate: no level of government gets to skip it.
American courts have read it to mean two things. Procedural due process is about fair steps, the how. Substantive due process is the more debated idea that some rights are so fundamental the government cannot take them no matter what process it follows.