Old English feohtan "to combat, contend with weapons, strive; attack; gain by fighting, win" (intransitive; class III strong verb; past tense feaht, past participle fohten), from Proto-Germanic *fe(u)hta (source also of Old High German fehtan, German fechten, Middle Dutch and Dutch vechten, Old Frisian fiuhta "to fight"), probably from PIE *pek- (2) "to comb, to pluck out" wool or hair (source also of Lithuanian pėšti"to pluck," Greek pekein "to comb, shear," pekos "fleece, wool;" Persian pashm "wool, down," Latin pectere "to comb," Sanskrit paksman- "eyebrows, hair"). Apparently the notion is "pulling roughly," or "to tear out one another's hair." But perhaps it is from the source of Latin pugnus "fist."
Spelling substitution of -gh- for a "hard H" sound was a Middle English scribal habit, especially before -t-. In some late Old English examples, the middle consonant was represented by a yogh. Among provincial early Modern English spellings, Wright lists faight, fate, fecht, feeght, feight, feit, feyght, feyt, feort, foight.
From c. 1200 as "offer resistance, struggle;" also "to quarrel, wrangle, create a disturbance." From late 14c. as "be in conflict." Transitive use from 1690s. To fight for "contest on behalf of" is from early 14c. To fight back "resist" is recorded from 1890. Well figt þat wel fligt ("he fights well that flies fast") was a Middle English proverb.