Origins
The word 'voice' descends from one of the most important PIE roots relating to human speech: *wekʷ- (to speak, to say). It entered English in the thirteenth century from Old French 'voiz' (voice, sound, word), from Latin 'vōx' (genitive 'vōcis'), meaning 'voice,' 'sound,' 'utterance,' 'word,' and 'tone.' The PIE root *wekʷ- and its derivative noun *wṓkʷs (voice, speech) produced cognates across the Indo-European world, making 'voice' one of the most rooted communication terms in the language.
Latin 'vōx' and its associated verb 'vocāre' (to call) generated a vast English vocabulary. 'Vocal' (pertaining to the voice) and 'vocalist' (a singer) come directly from 'vōcālis' (of or pertaining to the voice). 'Vowel' comes from Old French 'vouel,' from Latin 'vōcālis littera' (a voiced letter) — a vowel is, literally, a letter that has voice, a sound produced with vibration of the vocal cords and an open vocal tract, as opposed to a consonant (from 'consonāns,' sounding together with), which requires some closure or obstruction.
'Vocation' (from Latin 'vocātiō,' a calling, a summons) originally meant a divine call to a religious life, then expanded — especially after the Protestant Reformation — to mean any calling or career. 'Avocation' (from 'avocātiō,' a calling away) originally meant a distraction from one's vocation; it now means a hobby or secondary pursuit. 'Invoke' (to call upon), 'evoke' (to call out from), 'provoke' (to call forth), 'revoke' (to call back), 'convoke' (to call together), and 'advocate' (to call to one's side, to speak for) all descend from 'vocāre' with different prefixes.
Proto-Indo-European Roots
The PIE root *wekʷ- also produced Greek 'épos' (ἔπος), meaning 'word,' 'speech,' or 'song,' which gave English 'epic' (from 'epikós,' pertaining to a spoken word or narrative poem) and 'epopee' (a literary epic). The connection between 'voice' and 'epic' is etymologically direct: both descend from the PIE root for speaking, and both reflect the ancient understanding that poetry was spoken, chanted, or sung — that literature was, before it was written, an act of voice.
The political and philosophical sense of 'voice' — meaning the right to express an opinion or the expression itself ('the voice of the people,' 'to voice a concern,' 'to give voice to the voiceless') — draws on a metaphor already present in Latin. Roman 'vōx populi' (the voice of the people) was a set phrase, and 'vōx' could mean 'opinion' or 'vote' as well as physical sound. The modern usage of 'voice' in phrases like 'voice in government' or 'voice vote' continues this ancient equation of physical sound with political agency — the idea that to speak is to participate, and to be silenced is to be excluded.
In linguistics, 'voice' has a technical meaning: the grammatical category that describes the relationship between the verb and its subject. Active voice (the subject acts: 'the dog bit the man') and passive voice (the subject receives the action: 'the man was bitten by the dog') are the two primary voices in English. The term 'voice' for this grammatical category comes from Latin 'vōx,' used by Roman grammarians to translate Greek 'diáthesis' (disposition, arrangement). The phonetic concept of 'voicing' — whether the vocal cords vibrate during the production of a sound — also uses 'voice' in its most literal sense: a voiced consonant (like /b/, /d/, /g/) is one produced with vocal cord vibration, while a voiceless consonant (like /p/, /t/, /k/) is produced without it.