The significance of Gregg v. Georgia is that it made the death penalty legal once again. The death penalty had been illegal for a few years, but it was reinstated by Gregg. The death penalty has come to be something of a hot topic in American politics. There has been a great deal of controversy over whether it should be abolished and what sorts of reforms should be made to its administration. These would not have existed had the Court kept the death penalty illegal in Gregg.
However, it seems likely that the issue of the death penalty would have become a major political issue even if Gregg had been decided differently. America is too conservative of a country (and was even more so at that time) with regard to crime and punishment to have simply allowed the death penalty to be banned permanently.
The main impact of Gregg v. The State of Georgia was in the U.S. Supreme Court's decision that capital punishment (i.e., the death penalty) was constitutional so long as the procedures involved in its execution did not violate the Eighth Amendment to the Constitution. In other words, states were free to impose the death penalty, but the methods used in carrying out that penalty could not constitute "cruel and unusual punishment." A major result of that ruling by the Court was the development by the states of the practice of lethal injection. While older forms of capital punishment -- mainly firing squad and hanging -- were deemed "cruel and unusual," other, more modern forms (and, it should be noted, that death by firing squad was practiced in the State of Utah as late as the 1977 execution of convicted murderer Gary Gilmore) were still in use, including the gas chamber and the use of electrocution (the "electric chair," still in use and still controversial for the failures at times of those vested with responsibility for carrying out the execution to actually kill rather than torture through burning condemned convicts strapped to faulty electric chairs).
Gregg v. The State of Georgia, by sanctioning capital punishment, allowed individual states to continue to execute prisoners convicted of particularly heinous crimes.