Protecting society from criminals so that people are safe e.g by putting them in prison or even execution. The Catholic Church supports this.
Retribution
Making criminals pay for their actions according to the seriousness of their crimes. Some Christians support this because it is seen as fair.
Deterrence
Putting other people off from committing a similar crime. E.g. If a potential criminal sees a criminal being executed, they might not commit a crime.
Reformation
Helping offenders to understand what they have done wrong and to choose not to do it again. This is a popular Christian view because this is how Jesus treated sinners eg the adulteress
Jesus to the adulterness: 'Go and leave your life of sin'
The ultimate goal of punishment is always to rehabilitate
The Bible teaches to protect the weak and vulnerable (Psalm 82:3)
Old Testament teachings on punishment
People could be put to death for a range of crimes including murder and blasphemy
'An eye for an eye, tooth for a tooth' (Exodus 21:24)
Some Christians use the 'eye for an eye' passage to support the death penalty
In the New Testament, Jesus updates the 'eye for an eye' passage and emphasises forgiveness
Jesus: 'I tell you, not seven times, but seventy seven times (Matthew 18:21-22)'