A more thorough explanation:

The death penalty is a punishment authorized by the state that involves executing an individual for a specific crime. It is also known as capital punishment. Congress and state legislatures can prescribe the death penalty for crimes considered capital offenses. The Supreme Court has ruled that the death penalty does not violate the Eighth Amendment's ban on cruel and unusual punishment, but the Eighth Amendment does shape certain procedural aspects regarding when a jury may use the death penalty and how it must be carried out.

• In 1972, the Supreme Court invalidated existing death penalty laws in Furman v. Georgia because they constituted cruel and unusual punishment in violation of the Eighth Amendment.
• In 1976, the Supreme Court upheld Georgia's new capital sentencing procedures in Gregg v. Georgia, reasoning that the Georgia rules reduced the problem of arbitrary application as seen in earlier statutes.
• In 2002, the Supreme Court decided in Atkins v. Virginia that executing intellectually/developmentally disabled criminals violates the ban on "cruel and unusual punishment" because their cognitive disability lessens the severity of the crime, and therefore renders the extraordinary penalty of death as disproportionately severe.

These examples illustrate the history and application of the death penalty in the United States. The first example shows how the Supreme Court invalidated existing death penalty laws in 1972, while the second example shows how the Court upheld Georgia's new capital sentencing procedures in 1976. The third example illustrates how the Supreme Court decided in 2002 that executing intellectually/developmentally disabled criminals violates the ban on "cruel and unusual punishment."