mid-13c., faucon, from Old French faucon "falcon" (12c.), from Late Latin falconem (nominative falco) "falcon" (source also of Old Spanish falcon, Portuguese falcão, Italian falcone, Old High German falcho, German Falke, Dutch valk), probably from Latin falx (genitive falcis) "curved blade, pruning hook, sickle, war-scythe" (see falcate); the bird said to be so called for the shape of its talons, legs, or beak, but also possibly from the shape of its spread wings.
The other theory is that the Latin bird name falx is of Germanic origin and means "gray bird" (from PIE root *pel- (1) "pale"), which is supported by the antiquity of the word in Germanic but opposed by those who point out that falconry by all evidences was imported from the East, and the Germans got it from the Romans, not the other way round.
Old English feallan (class VII strong verb; past tense feoll, past participle feallen) "to drop from a height; fail, decay, die," from Proto-Germanic *fallanan (source also of Old Frisian falla, Old Saxon fallan, Dutch vallen, Old Norse falla, Old High German fallan, German fallen, absent in Gothic).
These are from PIE root *pol- "to fall" (source also of Armenian p'ul "downfall," Lithuanian puolu, pulti "to fall," Old Prussian aupallai "finds," literally "falls upon").
Meaning "come suddenly to the ground" is from late Old English. Of darkness, night, from c. 1600; of land sloping from 1570s; of prices from 1570s. Of empires, governments, etc., from c. 1200. Of the face or countenance from late 14c. Meaning "to be reduced" (as temperature) is from 1650s. Meaning "die in battle" is from 1570s. Meaning "to pass casually (into some condition)" is from early 13c.
To fall in "take place or position" is from 1751. To fall in love is attested from 1520s; to fall asleep is late 14c. To fall down is early 13c. (a-dun follon); to fall behind is from 1856. Fall through "fail, come to nothing" is from 1781. To fall for something is from 1903.
To fall out is by mid-13c. in a literal sense; military use is from 1832. Meaning "have a disagreement, begin to quarrel" is attested from 1560s (to fall out with "quarrel with" is from late 15c.).
c. 1200, "a falling to the ground; a dropping from a height, a descent from a higher to a lower position (as by gravity); a collapsing of a building," from Proto-Germanic *falliz, from the source of fall (v.). Old English noun fealle meant "snare, trap."
Of the coming of night from 1650s. Meaning "downward direction of a surface" is from 1560s, of a value from 1550s. Theological sense, "a succumbing to sin or temptation" (especially of Adam and Eve) is from early 13c.
The sense of "autumn" (now only in U.S. but formerly common in England) is by 1660s, short for fall of the leaf (1540s). Meaning "cascade, waterfall" is from 1570s (often plural, falls, when the descent is in stages; fall of water is attested from mid-15c.). The wrestling sense is from 1550s. Of a city under siege, etc., 1580s. Fall guy is attested by 1906.