An analgesic is a medicine that takes away physical pain. If you ask for pain relief, and the nurse says "Here's an analgesic," she's not trying to worsen your headache with a difficult word; she's just giving you a painkiller.
Breaking apart the word analgesic helps with pronunciation, ann-ull-JEE-zick. This isn't a proper root-word study, but the last three letters of the word look like "sick." If you're sick and have some pain and discomfort, you might get relief from an analgesic. As a noun, analgesic is the actual medicine, and as an adjective, it describes the effect of the medicine — a pill will have an analgesic effect, relieving the pain — unless it's really bad pain, in which case you'll need two.