A vow of silence or oath of silence was an oath made by an individual to not speak at all, or only to a limited number of people. The vow could be personal or religious in nature.
In 2369 after a Hunter found a Tosk in a holding cell aboard Deep Space 9 he asked him how he managed to get the help of these aliens without violating his oath of silence. (DS9: "Captive Pursuit")
Hupyrian servants took partial vows of silence which forbade them from speaking to anyone other than to their Ferengi masters. (DS9: "Prophet Motive")
Tom Paris threatened to send The Doctor's character in the Fair Haven holoprogram, Father Mulligan, to a nearby monastery with a vow of silence. (VOY: "Fair Haven")
A deleted scene from Star Trek, reveals that Nero had taken a vow of silence following the destruction of his homeworld Romulus.