The term Avox contains the Latin root vox, which means "voice" (Numen: The Latin Lexicon). The term also contains the Greek prefix a, which means "without" ("English Language Roots"). Hence, together, the word avox means without voice. The term perfectly describes the Avoxes, who are thought of as criminals and used as slaves.
In Chapter 6 of Suzanne Collins's The Hunger Games, we learn that an Avox is a person who has been charged as a traitor to the Capitol. The tongues of Avoxes are cut off so that they can't speak out against the Capitol.
We learn these details about Avoxes when Katniss recognizes the girl serving them at the Training Center, with her "dark red hair, the striking features, the porcelain white skin." However, Haymitch insists Katniss must be mistaking her for someone else for Katniss can't possibly know an Avox. Later, in her confidence to Peeta, Katniss reveals that she had first seen the girl with a boy while Katniss had been out hunting with Gale in the forests of District 12. The girl and boy had run past them in the woods, looking like they were running for their lives. The girl had called out for help, but Katniss had not responded and had watched the hovercraft appear out of nowhere, kill the boy, and whisk the girl away. Katniss feels very guilty for her behavior. Later, Katniss gets her chance to apologize to the girl, saying, "I should have tried to save you," but the girl only shakes her head and indicates that trying to save her would have only resulted in Katniss being turned into an Avox too.