Death Penalty or capital punishment has been exercised in all parts of the world since ancient times. It is as old as the beginning of history. It was part of the Justice system of ancient tribes. It has been followed in all religions and races. None of the historical records show any signs of who invented this punishment. You can find more details on it at the link below: en.wikipedia.org
As Aicha stated, the death penalty had been around since ancient times. The general assumption is, by imposing the ultimate punishment, it will discourage others from replicating the offending behavior. In the last few decades, state government here and abroad had ceased capital punishment because of two arguments. The first, is that it contradicts Christian ideals. Jesus and the Bible had promoted the ideal of forgiveness. The USA can be seen as a hypocrite to allow capital punishment to exist. We claim ourselves to be God fearing, and virtuous people, then contradict the ideals of mercy and forgiveness by taking a human life. Some will argue that what is "victim's right for justice," is actually an act of retribution. The 2nd argument against capital punishment are the crime statistics data. England, up until they had ceased the death penalty, had staged DAILY public executions. Instead of deterring the unwanted behavior, the public became desensitized to displays of public hangings. Public executions became a form of entertainment. People would watch the hangings, and laugh on the condemned. People would make insensitive remarks on the condemned to elicit a laughing response from the general audience. The crime statistic proved public displays of executions did not deter people from committing crimes. England realized that the lack of jobs and economic opportunities are the catalyst to criminal behavior.