Execution by Beheading

"[The sin] shall surely be avenged"Exodus 21:20.

Transgressors of certain sins are to be executed via decapitation.

Execution by Strangulation

"He shall be put to death"Exodus 21:16.

Transgressors of certain sins are to be executed via strangulation.

Hanging

"And you shall hang him on a wood[en pole]"Deuteronomy 21:22.

We are commanded to hang the corpses of certain criminals executed by the courts. (See Negative Commandment 66.)

Burying the Dead

"You shall surely bury him on the same day"Deuteronomy 21:23.

We are commanded to bury those executed by the courts on the same day of their execution.

The same applies to all dead—every Jewish deceased must be buried on the day of the passing. (This is why a person who died with no one to care for his interment is called a "met mitzvah" (a "mitzvah corpse"), because it is a mitzvah on everyone to ensure that he is buried.)

Leaving a Corpse Hanging Overnight

"His body shall not remain all night upon the tree"Deuteronomy 21:23.

It is forbidden to leave a corpse that was hung by the courts hanging overnight.

This because only two offenses are punishable by hanging [post execution]: idolatry and blaspheme. As such, someone seeing the hanging corpse might contemplate the reason why the individual was hanged, and will come to blaspheme G‑d in his thoughts.

[This prohibition also includes leaving any Jewish corpse unburied overnight, unless the delay in the internment is in honor of the deceased; e.g., to allow time for relatives to arrive for the funeral.]