The process, as I understand it, goes like this
Dwarf gets injured, goes to the hospital and is designated for diagnosis.
The diagnoser looks at all the tissue layers on the dwarf and sees which ones are torn, broken and such, as well as any other general conditions (syndromes, infections etc)
Every injury needs cleaning. A single 'clean patient' job will take care of all wounds at once.
Torn 'soft' tissues generally need sutures and bandages. They won't affect a dwarf's work otherwise though, unless nerves or ligaments were severed.
Broken bones need to be set. Simple fractures only require a 'set bones' job, but shattered bones and compound fractures require some surgery and a traction bench. (Not entirely sure about that part. Correct me if I'm work)
Limbs with broken bones will require a splint or a cast if they want to remain functional while they heal. (This is the best part of the new health care IMO)
Necrosis (rotting tissue) is usually cured by surgery cutting away the affected tissue.
Severed limbs, strangely enough, don't seem to require ANY treatment, since the game doesn't 'see' an injury.
The one part I'm not sure about is damage to the central nervous system. I don't think I've ever seen a brain injury that wasn't fatal, but I'd imagine my dwarves aren't very good brain surgeons. I think I've seen them operate on the spine, but I don't think it helped much.
Syndromes and associated conditions currently don't have any treatment. So your dwarves are stuck being nauseous, feverish or in pain until they get over it.
Just now I kinda realized I went a bit off-topic from the OP's question. I know that in 40d, every season, the game did a check on every injured dwarf to determine if they healed up a step or not. In the new version, it works just as Girlinhat said. As I understand it, the number denotes the amount of time it takes to heal. I don't think anybody ever figured out what those numbers mean exactly, just that lower is faster.