Origins
The verb 'sue' entered English around 1200 from Anglo-Norman 'suer,' from Old French 'sivre' (later 'suivre'), meaning 'to follow, to pursue,' from Vulgar Latin '*sequere,' from Classical Latin 'sequī' (to follow), from PIE *sekw- (to follow). The word is a spectacular example of phonological erosion: Latin 'sequī' (two syllables) became Vulgar Latin '*sequere' (three syllables), then Old French 'sivre' (two syllables), then Anglo-Norman 'suer' (two syllables), and finally English 'sue' (one syllable) — with the original meaning of 'following' preserved throughout.
In Middle English, 'sue' retained the broad French sense of 'to follow' in addition to its legal meaning. 'To sue for peace' meant to follow (pursue) peace; 'to sue for someone's hand' meant to follow them romantically. The legal sense — to follow someone through the courts — gradually dominated, and by Modern English 'sue' is primarily a legal term.
Latin Roots
The word generated a remarkable cluster of English derivatives, all from French forms of Latin 'sequī.' A 'suit' was originally a following or pursuit — hence both 'lawsuit' (a legal following) and 'suit of clothes' (a set of garments that follow/match each other). A 'suite' is a set of things that follow together: hotel rooms, musical movements, software programs. A 'suitor' is one who follows — whether in courtship or in court. 'Ensue' (to follow as a result) preserves the French form of 'in-' + 'sequī.'
The verb 'pursue' — from Old French 'porsivre,' an alteration of 'prosivre,' from Latin 'prōsequī' (to follow forward) — is a close relative. Where 'sue' narrowed to legal following, 'pursue' retained the broader sense of chasing or striving after. Together, they demonstrate how a single Latin verb can split into multiple English words with distinct but related meanings.