Similar to how plants and (in some cases) animals were used to heal ancient people in our own world; Opium poppies (from which comes morphine), Honey (which helps sooth sore throats), marigold (I think this helps heal infections, if I'm wrong let me know), leeches (blood letting, not successful but still used) and maggots. There are freshwater and salt water variants of the merpeople, so answer involving salt and fresh water life would be helpful. My merpeople live in medium and shallow levels of water but can go to deeper areas if they need to. So for example if they were bitten by a shark, 'stepped' on a sea urchin, or cut on coral, what underwater life would be used to aid the injured in a merperson culture?
If you're looking for detailed descriptions of individual treatment then you will need to make the question more specific but in general underwater medicine would not be very different to surface medicine.
There would be a number of advantages, for example moving patients around would be much easier when they are already floating in water. There would also be some difficulties, for example mixing up and grinding up medicines without them getting carried away by the water would be harder. You might see mixing bowls replaced by flexible bags for example.
Medicines would come from the same place ours did - plants and minerals that have useful properties and then finding ways to extract and preserve those properties.
Surgery and similar treatments would act in a very similar way, although keeping things clean would be harder and blood would tend to form a cloud making it hard to see.
Assuming they would react to drugs the same way a human would a guide to the common edible and medicinal sea plants could help.
And what about NaCl liquid?It is like sea water solution used for washing out the wound and better healing (NaCl) = 0,9%)
For medical purposes, saline is often used to flush wounds and skin abrasions. Normal saline will not burn or sting when applied- Saline is also often used for nasal washes to relieve some of the symptoms of the common cold.
Here's a link for some sea salt remedies http://www.earthclinic.com/remedies/sea_salt.html