Old English muþ "oral opening of an animal or human; opening of anything, door, gate," from Proto-Germanic *muntha- (source also of Old Saxon, Old Frisian muth, Old Norse munnr, Danish mund, Middle Dutch mont, Dutch mond, Old High German mund, German Mund, Gothic munþs "mouth"), with characteristic loss of nasal consonant in Old English (compare tooth), probably an IE word, but the exact etymology is disputed. Perhaps from the source of Latin mentum "chin" (from PIE root *men- (2) "to project," on the notion of "projecting body part"), presuming a semantic shift from "chin" to "mouth."
In the sense of "outfall of a river" it is attested from late Old English; as the opening of anything with capacity (a bottle, cave, etc.) it is recorded from mid-13c. Mouth-organ attested from 1660s. Mouth-breather is by 1883. Mouth-to-mouth "involving contact of one person's mouth with another's" is from 1909.
Word of mouth "spoken words, oral communication" (as distinguished from written words) is by 1550s. To put words in (someone's) mouth "represent as having said what one did not say" is from late 14c.; to take the words out of (someone's) mouth "anticipate what one is about to say" is from 1520s. To be down in the mouth "dejected" (1640s) is from the notion of having the corners of the mouth turned downward.