Adding more than you probably want to know to what Sue said, my criminal law textbook (Criminal Law and Its Processes, edited by Kadish and Schulhofer) explains it more or less like this:
In 1972, in “Furman v. Georgia”, the U.S. Supreme Court held by a 5-4 majority that capital punishment as it was being administered at the the time was a violation of the Eighth Amendment’s prohibition on cruel and unusual punishment. Two of the justices (Brennan and Marshall) held that all capital punishment was unconstitutional. Three others (Douglas, Stewart, and White) took the position that it was the erratic and unpredictable nature of imposition of the death penalty that was unconstitutional. The Chief Justice, along with Blackmun, Powell, and Rehnquist, dissented, stressing the long tradition of capital punishment in the U.S.
It was not, however, a complete and utter rejection of capital punishment. States that wanted to retain the death penalty took some time to reformulate their legislation (doing stuff like making the death penalty mandatory in certain cases) and establish clearer guidelines as to who should get death. By 1976, the U.S. Congress and at least 35 states had taken this action, effectively “legalizing” the death penalty again.
There again, I think you can make a pretty good case for the imposition of the death penalty still being pretty arbitraty in terms of race, income, etc., but that’s probably better suited to another forum.