I thought that I might have covered this before, but nope. I have written about crime but not specifically about capital punishment. I had to reread my post titled "Kill the criminals" to see for sure, but I tend not to be that obvious in my titles. That is good I haven't written about it, because it gives me a topic. I have been thinking about it over the past couple of days with the announcement of the execution of the Washington sniper. It is interesting, I think, and it is also a topic that everyone seems to have an opinion on (including myself). I will touch more on that comment in the "How I relate" section.
But let's get started, shall we?

Eye for an eye: This is the basic argument in support of capital punishment. If you take someone else's life, your are then not entitled to your own. This also satisfies a basic human desire for revenge, which, while not officially part of the justice system, is a practical part of it. It is often argued as well that some crimes are so horrific but do not involve the death of the victim that capital punishment of criminal is justified in these situations as well. It still falls under the "eye for an eye" category because the damage to the victim, while not death, can be effectively death; the crime does not allow the victim to live the life they were entitled to live.
The second main support of capital punishment is the safety of society. This takes places in two ways: the actual criminal is killed, and thus unable to harm others; and the fear of death will deter others from committing crimes that will result in their own death. Both of these points are fairly self explanatory.

A third support, while lesser, is the cost basis - it costs more to imprison someone that to execute them. Society has already borne the cost of the crime - now they have to bear the financial cost of feeding, housing, clothing, and possibly rehabilitating the criminal. Since the criminal did not give these options to the victim, why should society have to continue to pay? It would be cheaper just to get rid of them.
But what if they are wrong? There are two strong arguments against capital punishment. Depending on who you listen to, the error possibility is the strongest. This position holds that if the wrong person is convicted - which has happened many times - and executed, it cannot be undone.

The second strong argument is that life is the most important thing. Basically, yes, the criminal has committed terrible acts and deserves to be punished, but the right to life is absolute. Lock them up, throw away the key, put them in solitary, fine - but let them live. This is connected to the idea that if a government condemns murder and then commits it, it is on shaky moral ground.
The cost argument has been used by the opposition as well. It holds, and I cannot verify this, that the cost to execute a person, when all the court costs and appeals are factored in, are greater than the cost to have the person live their life in prison. It is a weaker argument, but it is almost always used as a refutation of the cost argument on the other side.
What I think: It is interesting to note that virtually all countries, even those that do not have actual capital punishment laws for criminal behaviour, still have it as a potential punishment for treason. My position is that I am opposed to it, but I don't have a problem with states (such as the US or China) that do have it. I don't believe the in deterrent effect -in fact I believe that the more violent the state is, the more violent the populace is. I understand it as a form of vengeance, and I don't have a problem with that. I do have a problem with the potential to kill innocent people, however, and I do believe that the punishment is not applied fairly or consistently. Nor do I think that most people, no matter the crime, deserve death; and I think attempts to differentiate the crimes that deserve death are pointless. To some people, having their life savings stolen may be the worst thing to happen to them and they may feel that the criminals deserve death, but that doesn't mean it is true. I am reminded of a Star trek episode where on one planet, the punishment for any crime, no matter how small, was death - and I liked that application of the law. Fair.
How I relate: This is what I was getting at earlier. Very few of us will ever be in the situation to be executed, or to even know someone who may be or has been executed. So I don't think that I really can relate to this. The common question people ask is a hypothetical one, along the lines of "If something happened to someone you love, wouldn't you want the person responsible to be punished?" and the answer, of course, is moot - it hasn't happened, and if it does, I will deal with it then. I do know, that no matter the outcome in that situation, nothing would make me feel better. I don't think "closure" would happen just because of an execution.
I had considered, at one point, about doing a post about the purpose of the criminal justice system, but I don't think that it is worth it. It could be rehabilitation, it could be punishment, it could be safety of society, it could be all three. And I think it is all three, but I don't think I can articulate that.