What is the philosophy of punishment?

In general, the philosophy of punishment covers these main ideas:

Deterrance: the idea that punishing someone for their actions will deter (make less likely) those same actions in the future, both by that person and by other people who know about the punishment given to another.

Retribution: the idea that punishment is needed to get revenge for bad actions.

Rehabilitation: the idea that a punishment can be affixed that will help the offender to change into a better person who no longer acts in the same detrimental ways.

Restoration: the idea that a punishment can be affixed that will help solve the problem that the offender created. This could be repayment of something that someone stole, or doing community service for someone who offended the community in another way. The idea of replacing good with bad and healing the harm that one has caused.

Different people embrace different ideas, so all of these don't come as a package, but those are the basic ideas behind punishment.