Pain is a common sensation that humans and presumably other animals react to when their nerves detect tissue damage or possible damage. What does and does not feel pain, and whether we should care about minimizing pain in anything other than humans (or which humans), is an ongoing question in ethics.
Further complicating the issue is the fact that pain is highly subjective. For example, a T-800 is unlikely to be fazed by buckshot rounds, while a human, when exposed to buckshot at a high velocity, may rate their pain at a solid "11" on the standard 1-10 scale.
 Pain and Survival
Not all behaviors are instinct. Pain and other unpleasant senses allow for an animal to learn from its environment; the spikey thing hurts my tongue, don't eat the spikey thing. The shiny orange bug tastes awful, don't eat shiny orange bugs. Any creature that ignores pain, such as a particularly ornery rabbit or Guy Fieri, may find themselves winning the Darwin Awards.
Painkillers are various type of pharmaceuticals (or sometimes, a placebo) that are administered to people to manage pain. Often these are opiates, so named because they derive from opium, and as such they may be highly addictive. Where's my morphine, doc?