"move air, produce a current of air," Old English blawan "to blow (of the wind, bellows, etc.), breathe, make an air current; kindle; inflate; sound (a wind instrument)" (class VII strong verb; past tense bleow, past participle blawen), from Proto-Germanic *blæ-anan (source of Old High German blaen, German blähen), from PIE root *bhle- "to blow."
Transitive sense of "carry by a wind or current of air" is from c. 1300; that of "to fill with air, inflate" is from late 14c. Of noses from 1530s; of electrical fuses from 1902. Meaning "to squander" (money) is from 1874; meaning "lose or bungle (an opportunity, etc.) is by 1943. Sense of "depart (some place) suddenly" is from 1902. For sexual sense, see blow-job.
As a colloquial imprecation by 1781, associated with sailors (as in Popeye's "well, blow me down!"); it has past participle blowed.
To blow (a candle, etc.) out "extinguish by a current of air" is from late 14c. To blow over "pass" is from 1610s, originally of storms. To blow hot and cold "vacillate" is from 1570s. To blow off steam (1837) is a figurative use from steam engines releasing pressure. Slang blow (someone or something) off "dismiss, ignore" is by 1986. To blow (someone's) mind was in use by 1967; there is a song title "Blow Your Mind" released in a 1965 Mirawood recording by a group called The Gas Company.