Most of the specific arguments against the death penalty (aside from the universal idea that it's wrong to kill people, and that the state shouldn't get to decide who does or does not deserve to live) are based on its use in the US, although I'm sure that many of them apply to other countries as well:
- It's not a deterrent: crime rates are equal to and sometimes higher than countries/states which do not have the death penalty, meaning that it doesn't do anything to prevent violent crime, which was a large part of the reason for introducing it.
- It's no less expensive than imprisonment: most offenders sentenced to the death penalty are kept on 'death row' for an extensive period of time, countering the common argument that the death penalty saves the state money in comparison to a life sentence.
- It's discriminatory: there is compelling evidence to suggest that a black person is more likely to be sentenced to death than a white person convicted of exactly the same crime. You're also more likely to receive the death penalty if you happen to be poor. There are people who view the death penalty as essentially being a form of state sanctioned eugenics.
- It can be a barrier to a trial: many countries will not extradite a defendant to a country where there is a risk that they will face the death penalty, which can prevent that person from being tried at all if the country seeking extradition refuses to guarantee that capital punishment won't be used.
It punishes the offender's family more than the offender themselves: if you subscribe to the view that retribution should be a principal focus of the justice system, you might resent that people who commit heinous crimes are given what might be seen as an easy way out. Indeed, there are a number people who would prefer to die than to be given a life sentence. Meanwhile, the offender's family and friends are made to face the grief of losing a loved one, while they themselves have done nothing wrong.
- It's pretty hypocritical: say, for example, that a person is being tried for murder, and immediately after the 'guilty' verdict is announced, a member of the jury picks up a gun and shoots the defendent dead. That juror would be arrested and tried for murder themselves. Yet, had that defendent been sentenced to death, they would have died anyway. Why, then, should government officials be above the law? What gives them the right to take the life of another, when the average citizen cannot?
- Public support is declining: the exact numbers vary, but national surveys in the US have found that public support for the death penalty has been consistently declining since at least the 1990s. The consensus is that very soon (or already, if you favour Pew Research's poll over Gallup's) the majority of Americans will not support the death penalty.