I'm familiar with jewish laws governing murder, but I can't recall ever having heard of jewish laws which govern assault.
What are some of the rules concerning classifying and punishing people who commit assault?
The Business Halacha Institute writes:
A person who inflicts physical injury is liable for a five-part payment: nezek (disability), tzaar (pain), ripuy (medical expenditures), sheves (lost wages), and boshes (embarrassment) — whichever are relevant (C.M. 420:3).
Nezek relates to the inherent, permanent disability of the injury, such as loss of limb or ability to see or hear. Tzaar is payment for the pain and suffering associated with the assault or injury, even if there is no disability. Ripuy covers medical expenditures associated with healing the injury. Sheves covers the temporary loss of wages while the victim is convalescing. [Permanent loss of work due to disability is included in nezek.] Boshes covers the embarrassment inflicted on the victim through injuring or hitting him. This payment is required only when the perpetrator intended to embarrass or injure (C.M. 421:1, 11).