Your body is full of nerve endings in your skin and tissues. Some of these nerve endings can sense pain, like from a burn or a blow to a body part. When cells in your body are injured or damaged, they release a chemical called prostaglandin.
When you take a pain killer medicine, it keeps injured or damaged cells from making and releasing prostaglandin. When the cells don't release this chemical, it means that the brain won't get the pain message as quickly or clearly. So your pain goes away or becomes less severe for as long as the cells aren't releasing the chemical. The pain killers you take will either disrupt the signal heading from the injured area to the brain, or it will change the way your brain interprets that signal.