Old English pyffan, *puffian "to blow with the mouth," of imitative origin. Compare pouf, from French. Especially "to blow with quick, intermittent blasts" (early 14c.). Meaning "pant, breathe hard and fast" is from late 14c.
The meaning "to fill, inflate, or expand with breath or air" is by 1530s. The intransitive sense, in reference to small swellings and round protuberances, is by 1725. The transitive figurative sense of "exalt" is from 1530s; shading by early 18c. into the meaning "praise with self-interest, give undue or servile praise to." Related: Puffed; puffing.
Old English (West Saxon) næddre (Mercian nedre, Northumbrian nedra), "a snake; the Serpent in the Garden of Eden," from Proto-Germanic *naethro "a snake" (source also of Old Norse naðra, Middle Dutch nadre, Old High German natra, German Natter, Gothic nadrs), from PIE root *nētr- "snake" (source also of Latin natrix "water snake" (the sense is probably by folk-association with nare "to swim"); Old Irish nathir, Welsh neidr "snake, serpent").
Since Middle English restricted to use as the common name of the viper, the only poisonous British reptile (not generally fatal to humans), then by extension applied to venomous or similar snakes elsewhere (puff-adder, etc.). The modern form represents a faulty separation 14c.-16c. of a nadder into an adder, for which see also apron, auger, nickname, orange, humble pie, aitchbone, umpire. Nedder is still a northern English dialect form. Folklore connection with deafness is via Psalms lviii.1-5. The adder is said to stop up its ears to avoid hearing the snake charmer called in to drive it away.