When you do something without saying a word, you do it mutely. If you can't carry a tune, you might sit mutely while everyone else sings "Happy Birthday" to your sister.
The adverb mutely is similar to silently, or "making no noise at all," except that it refers specifically to speech. If you don't talk or vocalize, you are mute, so if you smile mutely at your best friend, you don't speak to her at all. Both mute and mutely come from the Latin mutus, "silent or speechless."