The brief order stayed the execution pending a writ of certiorari to the court because of the urgency of Bucklew's pending execution. Four justices dissented from the order: Chief Justice John Roberts, and Justices Clarence Thomas, Samuel Alito and Neil Gorsuch.
Bucklew contends [SCOTUSblog report] that execution by lethal injection would violate the Eighth Amendment's ban on cruel and unusual punishment, because he suffers from a rare disease that would be exacerbated by the method of execution. The disease has caused "unstable, blood-filled tumors to grow in his head, neck, and throat," and Buckley posits that the execution will cause him a "gruesome and painful" death.
Bucklew was convicted [procedural history, PDF] by a Missouri state court jury of murder, kidnapping and rape. The jury recommended, and the trial court subsequently sentenced Bucklew to death.