When combined, the meaning is crystal clear: 止 (stop) + 痛 (pain) + 药 (medicine) = medicine that stops pain.
The term 止痛药 itself is a direct translation of a medical concept, but its use is embedded in China's dual medical system. In China, there's a strong distinction between:
For an English speaker, the main cultural difference is in the pharmacy (药店, yàodiàn). In the West, you might grab a bottle of Tylenol or Advil off the shelf without speaking to anyone. In China, while over-the-counter (非处方药, fēichǔfāngyào) painkillers are readily available, it's very common to approach the counter and describe your symptoms to the pharmacist, who then recommends a product. They act more as a first-line consultant. So, knowing how to say “I need a painkiller for a headache” is a very practical skill.
止痛药 is a neutral, standard term used in all contexts, from casual conversation to a formal doctor's consultation.
It's a purely functional term without strong positive or negative connotations, though like in English, the idea of being dependent on them can be viewed negatively.