Old English fæger "pleasing to the sight (of persons and body features, also of objects, places, etc.); beautiful, handsome, attractive," of weather, "bright, clear, pleasant; not rainy," also in late Old English "morally good," from Proto-Germanic *fagraz (source also of Old Saxon fagar, Old Norse fagr, Swedish fager, Old High German fagar "beautiful," Gothic fagrs "fit"), perhaps from PIE *pek- (1) "to make pretty" (source also of Lithuanian puošiu "I decorate").
The meaning in reference to weather preserves the oldest sense "suitable, agreeable" (opposed to foul (adj.)). Of the main modern senses of the word, that of "light of complexion or color of hair and eyes, not dusky or sallow" (of persons) is from c. 1200, faire, contrasted to browne and reflecting tastes in beauty. From early 13c. as "according with propriety; according with justice," hence "equitable, impartial, just, free from bias" (mid-14c.).
Of wind, "not excessive; favorable for a ship's passage," from late 14c. Of handwriting from 1690s. From c. 1300 as "promising good fortune, auspicious." Also from c. 1300 as "above average, considerable, sizable." From 1860 as "comparatively good."
The sporting senses (fair ball, fair catch, etc.) began to appear in 1856. Fair play is from 1590s but not originally in sports. Fair-haired in the figurative sense of "darling, favorite" is from 1909. First record of fair-weather friends is from 1736 (in a letter from Pope published that year, written in 1730). The fair sex "women" is from 1660s, from the "beautiful" sense (fair as a noun meaning "a woman" is from early 15c.). Fair game "legitimate target" is from 1776, from hunting.
Others, who have not gone to such a height of audacious wickedness, have yet considered common prostitutes as fair game, which they might pursue without restraint. ["Advice from a Father to a Son, Just Entered into the Army and about to Go Abroad into Action," London, 1776]