Old English æppel "apple; any kind of fruit; fruit in general," from Proto-Germanic *ap(a)laz (cf. Old Saxon, Old Frisian, Dutch appel, Old Norse eple, Old High German apful, German Apfel), from PIE *ab(e)l "apple" (cf. Gaulish avallo "fruit;" Old Irish ubull, Lithuanian obuolys, Old Church Slavonic jabloko "apple"), but the exact relation and original sense of these is uncertain (cf. melon).
In Middle English and as late as 17c., it was a generic term for all fruit other than berries but including nuts (e.g. Old English fingeræppla "dates," literally "finger-apples;" Middle English appel of paradis "banana," c.1400). Hence its grafting onto the unnamed "fruit of the forbidden tree" in Genesis. Cucumbers, in one Old English work, are eorþæppla, literally "earth-apples" (cf. French pomme de terre "potato," literally "earth-apple;" see also melon). French pomme is from Latin pomum "apple; fruit" (see Pomona).
Apple of Discord (c.1400) was thrown into the wedding of Thetis and Peleus by Eris (goddess of chaos and discord), who had not been invited, and inscribed kallisti "To the Prettiest One." Paris, elected to choose which goddess should have it, gave it to Aphrodite, offending Hera and Athene, with consequences of the Trojan War, etc.
Apple of one's eye (Old English), symbol of what is most cherished, was the pupil, supposed to be a globular solid body. Apple-polisher "one who curries favor" first attested 1928 in student slang. The image of something that upsets the apple cart is attested from 1788. Road apple "horse dropping" is from 1942.