What are some important elements of "due process of law"?

Due process of law is a constitutional construct that stands for the proposition that the government cannot take "life, liberty, or property" without providing procedural protections to the person the government seeks to deprive. In other words, we cannot sentence someone, imprison someone, or take away someone's property without following certain procedures. At the very least, we would perceive these actions to be unfair.

In the Fifth Amendment of the Constitution, the protections named are the right to an indictment, the right against double jeopardy, and the right against self-incrimination.  These ensure a process before someone can be prosecuted, a protection against being tried for the same crime again after having been acquitted of that crime, and a right not to have to take the stand in a proceeding against oneself. 

In the Sixth Amendment, we have the right to a public trial, a speedy trial, and an impartial jury. A public trial implies that people will be present to ensure the government follows all the rules, and a speedy trial is a right that exists so the government cannot simply put you in jail, throw away the key, and forget about you. An impartial jury is meant to stop the government from rounding up all your enemies and asking them to judge you. Also in the Sixth Amendment are the right to be informed of what you are being charged with, which is not something all countries do, believe it or not. The Sixth Amendment also requires that you can cross-examine any witnesses against you, and this is a very important right, too. The right to subpoena witnesses to testify for you is part of this amendment, as is the right to counsel. If you do not have the power to call witnesses or an attorney, it is just you against the government, which is an unfair contest. 

These are all quite important rights for a defendant and not available in many countries in the world. If you are going to be executed, incarcerated, or fined, then at least you know you had these protections going through the process.