From Old English 'bītan,' PIE *bʰeyd- (to split, to cleave) — the same root behind 'bit,' 'bitter,' and 'beetle.'
To use the teeth to cut into or grip something; to sting or cause a sharp pain.
From Proto-Indo-European *bheyd- ("to split, to pierce"), through Proto-Germanic *bitanam ("to bite, to cut") and Old English bitan ("to bite, sting, cut"). The PIE root *bheyd- (or *bhid-) conveys the action of splitting by pressure — the same root yields Latin findere ("to split, cleave") via *bhid-, Sanskrit bhinatti ("he splits"), and English fission (through Latin). The Proto-Germanic strong verb *bitanam had the ablaut series *bait-/*bit-/*bitanam, giving Modern English bite/bit/bitten. Old English bitan -> Middle English biten -> Modern English bite. In Old Norse the cognate bita meant both the act
A 'bit' (a small piece) is literally 'something bitten off.' The 'bit' of a horse's bridle is what the horse bites down on. And 'bitter' originally described the biting, sharp taste — the taste that makes you feel as if something has bitten your tongue. Even 'beetle' comes from this root: Old English 'bitela' meant 'the biter.'