A minyan is a gathering of ten male Jews of sound mind and responsibility to join together for matters of sanctity. A minor, who is not yet of full sound mind and competence, does not count as part of the minyan. When he reaches the age at which he is obligated to fulfill the mitzvot (bar mitzvah), he is counted as part of the minyan.
There are some Rishonim who maintain that in extenuating circumstances (b’sha’at hadchak), nine adults may include a minor who is holding a Chumash in his hand as part of their minyan. However, in the opinion of most poskim, even in extenuating circumstances, a minor may not be considered part of the minyan, and that is how we practice. Nevertheless, in a situation in which the minyan will be canceled completely unless he is counted, possibly causing some of the members to distance themselves from Judaism, he can be counted as part of the minyan.
Since a person who is deaf-mute does not have a way of communicating with the world, the Chachamim say that he is considered a shoteh and he is exempt from fulfilling the mitzvot. Therefore, he may not be counted as part of the minyan (Chagigah 2b; Shulchan Aruch 55:8). The poskim disagree as to whether or not the status of a deaf-mute person changes, if he was taught to communicate with his surroundings, either by sign language or by reading and writing. Since joining a minyan is a rabbinic mitzvah, the halachah follows those poskim who are lenient, and therefore he may be counted in a minyan.
The definition of an adult is one who has reached thirteen years of age and two hairs have grown in. Yet, practically we are not strict about this because praying in a minyan is a rabbinic obligation and we rely on the presumption that he has two hairs. Even if in actuality he does not, perhaps he did and they shed. See Mishnah Berurah 55, passages 31 and 40.