One of the main reasons people support the death penalty is that the criminal did something so horrendous that s/he deserves nothing better than death. This is commonly referred to as the "eye for an eye" argument. If a criminal takes someone's life, shouldn't his/her life be taken as well?
Many people will argue that, with the advancement in DNA testing and other crime scene invesitagtion techniques, many people who have been wrongly sentences have been proven innocent. How can we send people to death with even the slightest bit of doubt? Death penalty advocates will argue that because of this new technology, we are more sure that ever of a person's guilt or innocence.
Other reasons include that the death penalty cuts down on overcrowding in prisons and can be less costly than keeping a criminal in prison for many years. Also, if a criminal knows that s/he may receive the death penalty, it could deter the person from committing a crime.