Questions about the morality and efficacy of the death penalty have been around longer than the United States. In the Bill of Rights of the U.S. Constitution, Amendment VIII (which borrows language from the English Bill of Rights of 1689) states this:
Excessive bail shall not be required, nor excessive fines imposed, nor cruel and unusual punishments inflicted.
As you can see, the "cruel and unusual" clause applies not only to the death penalty, but to all types of legal punishment. Sentencing a person to death as punishment is not unusual — governments and leaders have been executing people for centuries — but in the United States, people question at what point a punishment becomes cruel or unusual.
Some people believe that taking another person's life — whether it's by a government or an individual — is by definition cruel and should therefore be banned. Those who don't oppose the death penalty see "cruel and unusual" in a different light, believing that as long as the person being executed doesn't suffer, it satisfies the Constitutional requirements.
In the United States, felons are normally executed in one of two ways: lethal injection or the electric chair, with lethal injection being the most common. It makes sense, too, because lethal injection inflicts the least pain, first putting the condemned to sleep, then stopping his breathing, and then stopping his heart.
Other execution methods have been used in the past, including the gas chamber, firing squad, and hanging.