What does Islam say about capital punishment?

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

Islam sanctions capital punishment if there is legal reason for it. For example, one who kills somebody intentionally should be killed if the relatives of the person who was killed do not pardon the killer or they refuse the blood money. This is called al-Qisaas in Islam. Also, a married person who commits Zina (adultery) should be stoned till death if his crime is proved by evidence or confession, and that it is not done by compulsion.

Killing bandits is a third example, since they rob the wealth of other people, frighten them, and even kill them as well.
Allah, the exalted, has enacted such punishment to safeguard lives and properties of people and to make them safe as well as to establish a safe society.

Allah Says (what means): {And there is for you in legal retribution [saving of] life, O you [people] of understanding, that you may become righteous.} [Quran 2:179]

So, al-Qisaas aims at safeguarding the others although it is a form of just killing; i.e. due to the killer's previous destruction of someone else. This is to give the society a lesson that prevents killers from doing such a crime again. Islam states that a person who betrays (a member in the society) should be humiliated due to his bad deed, whereas an upright one should be respected and honored.

Know that Allah has enacted forgiveness for the relatives of the killed person and urged them to do so if they like, with or without compensation.

Allah Says (what means): {…But whoever overlooks from his brother anything, then there should be a suitable follow-up and payment to him with good conduct. This is an alleviation from your Lord and a mercy...} [Quran 2:179]

Great is He, The Creator, The Maker, The King who knows and enacts what He wishes and what is good for His slaves.

Allah knows best.