There are different types of silent protagonists that have appeared in many different games.
See also Silent Protagonists Who Talk in Sequels.
Mutes are characters who are recognized within the game world as people who are incapable of speaking. Claude Speed from Grand Theft Auto 3 is considered a "mute." Characters who, in general, cannot speak are not considered "mutes." For instance, the Space Marine from Doom would be considered neither a mute nor a silent protagonist, since he is never spoken to, so it's unknown whether or not he can speak. This page deals with the concept of mute characters exclusively.
Secretive silent protagonists are a step up from mutes in complexity, in that they are essentially predefined, but are left with space for the player to insert his or her own thoughts and motivations, simply left unsaid and thus "secret". Secretive protagonists are almost never seen or heard speaking, but are treated by other characters as being able to speak, or having spoken in past, unseen events. The secretive protagonist's personality is defined indirectly, implied by the actions and reactions of the NPCs around him.
Secretive protagonists, therefore, can exist in an entirely linear game without compromising the ability of the player to self-insert. They can also possess background information, even a personal history, and still remain "open" for the player to establish a personal connection. For example, Gordon Freeman of Half Life 1 and Half Life 2's only experience with a weapon before the incident at Black Mesa was the use of a vacuum-powered tennis ball launcher, and in terms of its effect on gameplay or Gordon himself, is little more than insignificant trivia. He never speaks a word during the gameplay, yet is treated by his companion Barney Calhoun as a drinking buddy and partner. Silent player characters in many Japanese-style RPGs, such as those of Suikoden V and Persona 4 are secretive in that they rarely or never directly assert themselves, yet are treated as the center of the game's storyline and as primary decision-maker.
A common quality of game scripts that involve secretive protagonists is that surrounding NPCs are often written as to never assume that the player has reacted in a given way. When Alyx of Half-Life 2: Episode 1 coins the term " Zombine", her character does not assume that Gordon laughed (or did not laugh), instead reacting to herself as having made a bad joke, regardless of whether or not the player (as Gordon) thought the joke was funny.
Reactive silent protagonists are protagonists whose actions are assumed by the game's scripting, in spite of having no lines or dialog options. The character technically speaks, and other characters in the world react to player as if they have spoken, but the words are never actually shown or heard. Link from The Legend of Zelda series is considered a "reactive" silent protagonist. In Ocarina of Time, Princess Zelda asks Link what his name is, and while no dialogue circle appears indicating that Link has spoken, Zelda replies "Link??? That seems familiar" implying that he has indeed said something. The "Commander" personality addressed in the FMV cutscenes of the Command & Conquer series can be considered as a reactive silent protagonist in that the character is directly addressed by the onscreen NPCs, his or her reactions are assumed automatically to agree with (or at least submit to) the demands and motivations of the NPCs. A reactive silent protagonist is essentially a prescripted character whose lines of dialog have been excised. Reactive silent protagonists are more complex than Mutes because the player is thought to have had a choice, but simply assumed to have selected a prescripted option.
Jack, protagonist of Bioshock, is initially thought of as a Reactive (or Secretive) silent protagonist, though events later on in the game dramatically subvert the archetype.