As a modern country, the leader of the free world, with democracy in action, and a nation founded as a Republic, it is a bit unusual, even strange, that capital punishment is still permitted and practiced within the United States, which definitely makes our country appear as an outlier to the balance of the civilized world, especially considering that the 8th Amendment protects us from "cruel and unusual punishments", in which the death penalty clearly is both cruel and unusual.
The most obvious reason to put someone to death is to preclude that individual from mounting a counter-offensive against a regime, against other individuals, and as a punishment for crimes that they have committed. However, when a government or a court enacts this reckoning, they have done so after a deliberate calculation, perhaps even with a show-trial, to set forth their reasons for the execution of said prisoner.
Another valid reason for capital punishment is to demonstrate to the population that certain crimes are punished with certain death, and consequently to make these executions public so as to set forth an example of what happens to those that disobey edicts.
Still the fact of the matter is, that civilization, has quite clearly become less blood-thirsty when it comes to the deliberate execution of its subjects, in which this form of punishment was meted out far and wide for hundreds of years, encompassing the lowest of the low to kings and queens, who were executed for being on the wrong side of important or politically sensitive issues.
However, there are a multitude of problems with executions. First off, they are irreversible, so that if you put to death an innocent man, or a man misunderstood, or a man of a real value or need, you cannot bring that man back to life. Additionally, if you execute people for frivolous reasons, the population as a whole, will rebel internally and possibly externally, because they will not recognize nor respect an agency performing such a deed. Finally, the taking of another man's life, is to certain degree aggrandizing to you, God-like powers, but unfortunately, this power is only the power to destroy life without the appropriate balance of bringing forth life, which means that this tool is solely a negative power, and will ultimately undercut your regime and its moral authority.
Fortunately, civilization has matured over the years, and we have grown in our understanding that how we treat those that are less fortunate than us, less able than us, less wise than us, is the true definition of our compassion to humanity. Today, there is a much better understanding that within all of us is a combination of both good and evil traits and actions. In most countries, the unnecessary taking of another man's life, cold, calculated, and final, is now seen as not necessary for justice to be served. That is not because we are weak, or that we don't seek justice for crimes committed against others, but because we recognize that within each man is a reflection of ourselves, and that that image should be respected.
The value of a human life is more appreciated today, that is the mark of maturity and of a more civilized and reasoned society.