Execution is where state authorities kill someone for having committed an extremely serious crime, usually treason or especially terrible murders. In most countries where the death penalty is still provided for by law, using it is an option available to the sentencing judge: even if the jury or judicial panel recommends the death penalty, the presiding judge still has the option to lock the convicted person in a prison for the rest of their life. A person whose job is to execute others is an executioner.

Beheading

Beheading means cutting the convict's head off. It is one of the oldest execution methods and already mentioned in the Bible. Beheading used to be the standard method of execution in Scandinavia and Germany. Commoners were usually beheaded with an axe and noblemen with a sword. A special device, like guillotine, can also be used, as in France. Nazi Germany used guillotine to execute criminal convicts, such as murderers.

Centuries ago many countries used beheading as an execution for important people, including England. In England, many noblemen and even some kings and queens have been beheaded. There, the prisoner would be led up the scaffold and usually say a speech. Then, he/she would be blindfolded and put his/her neck onto a block. Then, the executioner would lift up his axe and swing it down onto the victim's neck. If the executioner was skilled and the axe was sharp, then the axe would usually cut through the bone and organs of the victim in one stroke. But if the executioner was inexperienced, then it may take many strokes before the head would be cut off.

Other ways of execution

Many countries do not allow executions as punishment any more, because it is too violent or immoral. However, many states of the United States and some other countries use it.  In the United States, less violent ways of execution are used than in the past. Here are some ways of executing people:

 Hanging: Using a rope to either break the convict's neck or to choke (or strangle) them.  Widely used around the world until the 20th century. Still today in use in some countries, such as Iraq, Singapore and Japan.
 Firing Squad:  Several people shoot and kill a person.  Armies around the world have long used this method, since guns and bullets are readily available.  Firing squad was the lawful means of execution in Finland until 1944, when death penalty was abolished (stopped by the law).  It was also used in the state of Utah in the 20th century.  In most cases, not all the shooters have real bullets.  After the execution, it is not possible to determine which of the people firing killed the person just executed.
 Gas chamber: killing a person by filling the air in a room with poison gas until the person cannot breathe and dies.  This method was used for executions in some U.S. states, and for mass murder by the Schutzstaffel during the Holocaust.

Lethal injection: Killing a person by placing poisons into their bloodstream. This is the choice of execution in most U.S. states that allow executions.
Electrocution: Killing a person by placing them in an electric chair and giving them a very high electric shock.

Old-fashioned methods 
Crucifixion: a person (or their corpse) is fixed to a timber by nails or by impalement. The Romans used crucifixion to punish traitors, rebels and runaway slaves because the Romans considered it the most unpleasant death. Death by this method may take days. Besides ancient Greece, ancient Rome, and the Persian empires, this method was also used in feudal Japan.
Drawing and Quartering:  A very violent form of execution common in Medieval and Renaissance Europe. It involved taking a person's organs out while he was still alive, then tying them to four horses going in different directions until their body is torn apart.  
"Breaking on the wheel", the executioner breaks all the bones of a person's limbs with a heavy object. The executioner wraps the person's limbs around a wheel from a carriage, and lifts the wheel to the top of a tall pole. Slowly, the person dies.
Crushing, also called pressing: used in the common law legal systems. A defendant who refused to plead ("stood mute") would be subjected to having heavier and heavier stones placed upon their chest until a plea was entered or the person suffocated.
Garroting, a method of strangulation used in Spain for hundreds of years.

Crime
Capital punishment