Most executions that occur in the U.S. are under state authority, I think, and each state has the right to do things its own way. Some don’t even enforce the death penalty [see graphic, red being states that permit executions, blue not]. Of those that execute their fellow citizens, they apparently have five legal methods to chose from. The most common is lethal injection. At least, it used to be. But the Danish manufacturer of a drug used in the deadly cocktail that first sedates, then paralyzes, and finally kills the condemned, has refused to allow its use in capital punishment. So states are searching for a new combination of drugs to do the trick.
A new two-drug cocktail was used in a recent execution in Ohio but it didn’t go very well as the the man (Dennis McGuire) took 25 minutes to die and appeared to suffer. That upset a lot of people — opponents of capital punishment, naturally. But even supporters who, while okay with a death penalty, want the execution carried out in a “humane” manner.
Then there’s hardcore righties who just shrug and say “So what?” They hold the view that no person sentenced to death has an automatic right to a “painless” execution and are even touting a return to firing squads.
You can read more on the issue here.