means A pungent root, related to mustard and cabbage, that's grated into a sharp, nose-clearing condiment.

from From 'horse' + 'radish,' the 'radish' part coming via Old English from Latin 'radix,' meaning 'root' — the very word 'radical' is a cousin. The 'horse' is the puzzler: in plant names 'horse' has long been slapped onto things to mean 'coarse, large, or strong,' the rough peasant version of the gentler radish. So 'horseradish' is essentially 'the brute radish' — though the folk imagination later spun tales of horses craving it, which is just a story the name invites.