Dun: Mayflies have two adult stages. They first emerge from the water as duns (scientifically known as the subimago stage). They then molt into the spinner (imago) stage, in which they mate and die. Sometimes the word "dun" is confusingly used to refer to a brownish gray color in fly tying materials.
Nymphs: The juvenile, underwater stages of mayflies, stoneflies, dragonflies, and damselflies and other aquatic insects whose juvenile stages are covered by hard exoskeletons. The word can also refer to the fishing flies which imitate these creatures, in which case it is used as a blanket term for flies imitating any underwater stage of an invertebrate (except for crayfish and leeches).
Spinner: There are two winged stages of adult mayflies. They emerge from the water as duns, molt on land (usually) into their fully mature stage, spinners. As spinners, they mate, lay eggs, and die.