These are the primary reasons I oppose capital punishment, i.e. the death penalty:
It assumes perfection on the part of the government. The judicial system, like any human enterprise, is flawed. Innocent people are regularly executed, even, occasionally, innocents who have presented compelling evidence of their innocence, evidence ignored by the courts. It is abundantly clear that government makes poor choices, suffers from widespread corruption, etc. Why should we trust it to decide who may live and who may not?
It is a waste of public resources. A capital trial, lengthy appeals process, and execution costs significantly more than a non-capital trial (or plea bargain) followed by lifetime imprisonment.
It is morally wrong. By executing a person, the state simply promotes the cycle of violence. This is inappropriate for a civilized society.
Criminal justice should be based on prevention, rehabilitation, and restitution, not punishment and revenge. Even for the most appalling crimes, life without parole should be the maximum consequence.
Obviously there are many more arguments; e.g., the death penalty does not deter crime, the death penalty does not promote the healing of victims and their families (instead prolonging and publicizing their suffering). However, I believe the "Big Three" above are so compelling and so well-supported that belief in any one of the three, despite rejection of all other arguments against capital punishment, should lead to an anti-capital-punishment position.
This is a brief position statement, so I have not included specific sources. However, I highly recommend that you read Dead Man Walking by Sister Helen Prejean, which addresses this subject in a persuasive and moving way.