Short answer: different people have different amount of active receptors.
In treatment, combination scores of Pharmacodynamics and Pharmacokinetics determine the final effect of the drug.
Receptors determine many effects of the drug in many pathways.
Different people also sense pains differently (Psychology).
The purpose of treatment is ...
I don't know of any interesting mechanism that is specific to pain killers, so I will instead answer for drugs in general.
Drug action is a complex process consisting of many steps. Let's take a simple example: A systemic direct inhibitor of a kinase. This drug would need to*:
Be absorbed into your bloodstream
Remain in your bloodstream for sufficient time
These are 'peak' values when administered via IV, with the exception of oxycodone. There will be some variation person to person depending on their opioid tolerance and any concomitantly administered medications.
Oxycodone: Not administered ...