The goal of your dog's pain treatment

In general, common drugs that are used in pain management fall into one of five categories: opioids, NSAIDs, α2-agonists, local anaesthetics, and "others".
In addition, multiple nutraceuticals (a food or naturally-occurring food supplement thought to have a beneficial effect on health) are available over the counter. Nutraceuticals include glucosamine, chondroitin and vitamin C.

A distinction can be made between analgesics and anaesthetics.  Analgesics (or painkillers) are used to relieve pain without eliminating sensation – in contrast to anaesthetics.

Overall, the choice of painkiller will depend on the type of pain (e.g. mild, moderate or severe pain) and on the individual dog (taking into account medical precautions, contraindications, adverse effects, a need for combination analgesia, etc.).

Pain relief may help dogs live longer because it reduces stress and increases their sense of well-being.

But be cautious, ibuprofen for example, causes the more side effects in dogs than it does in people. It can cause gastrointestinal upsets and ulcers as well as kidney damage.
Always ask your vet before using any drug intended for humans on your pet. Your vet can recommend an appropriate pain reliever.