There are many points in the death penalty debate, and which are considered primary depends on the people speaking and the context of the argument. What all arguments boil down to, however, is that some believe that killing another human being is wrong in all circumstances, while others believe that doing so is permissible in certain contexts. There are also, of course, points in between these two extremes that may land a person on one side of the debate or the other. In addition to the broad basic points, there are additional issues that are often discussed as well.
One main point in the death penalty debate is that capital punishment disproportionately punishes minorities and poor people because these are the people most likely to be sentenced to death. People in favor of the death penalty often deflect this criticism by talking about the issue in terms of cultures of violence rather than in terms of race. Either side may be able to find statistics supporting completely opposite views in this aspect of the death penalty debate.
Another point is that taxpayers do not have an obligation to support a life of leisure for prisoners who are interred for life sentences. People who are against the death penalty typically argue against this point by stating that prison is not a life of leisure and that in some cases prison is worse than death. Which perspective is more convincing depends on the person's view of the purpose of prison. Rehabilitative theories of prisons are certainly opposed to the death penalty, but people who believe that prisons are solely for punishment may be in favor.
In many cases, the death penalty debate revolves around whether or not capital punishment deters future crimes. There is compelling evidence on both sides of the debate on this issue, but the results often depend on the area in which the death penalty is used. Commonly, the issue of whether innocent people who have been wrongly convicted of a crime could be executed comes up as well. Some people believe that the debate should not hinge on the idea that the justice system is fallible, but others believe that it is better to play it safe when it comes to life and death.
There are many more nuanced and subtle points within the death penalty debate, but these are often elaborations on the two broad perspectives that killing is wrong or that killing is sometimes right. Justifications of either viewpoint can be made in philosophical, economic, or even religious explanations. It is important to remember that the death penalty debate is not only an academic exercise, but also a debate in which real peoples' lives are at stake.