the mouth: the opening in the face used to eat and drink and above all to speak, and so, by a natural step, the speech, words, or command that come out of it
This is the everyday meaning of the word and the root of every other one. It names the physical mouth of a person or animal, the part of the face that takes in food and gives out sound.
Because the mouth is where words come from, Hebrew lets the same word stand for the speaking itself. What a person says can simply be called their mouth, and to put words in someone's mouth is to tell them what to say. The most weighted form of this is the mouth of the LORD, which does not picture a body part at all but means God's own spoken word, command, or promise. A prophet can even be called someone's mouth: Aaron speaks for Moses the way a mouth speaks for the person behind it.
Most occurrences of the word fall on the speaking side rather than the anatomical one, because the Bible cares far more about what people and God say than about the organ itself. The senses below run along that single line, from the physical mouth to the words it produces.