The Shulhan Aruch (Orah Haim 186:2) rules that a child who has yet to reach the age of Misva obligation must nevertheless recite Birkat Ha'mazon after eating bread due to the parents' requirement of "Hinuch" – training their children in Misva observance. A child from the age of six must therefore be trained to recite Birkat Ha'mazon after eating bread.
This obligation, however, applies only on the level of "Mi'de'rabbanan" – Rabbinic enactment. As far as Torah law is concerned, a child is not technically required to recite Birkat Ha'mazon until he or she reaches the age of Misva obligation (thirteen for a boy, twelve for a girl). This lower status of obligation precludes the possibility of a child reciting Birkat Ha'mazon on behalf of an adult in many cases. In situations where an adult bears a Torah obligation to recite Birkat Ha'mazon, Halacha would not allow a child, who is included in this obligation only by force of Rabbinic enactment, to recite Birkat Ha'mazon on the adult's behalf. Since the adult and the child bear different levels of obligation, the child's recitation is ineffective in satisfying the adult's requirement.
This Halacha would not apply, however, in cases where the adult and child share the same level of obligation. If the adult partook of a Ke'zayit of bread but did not eat "Kede Sevi'a" – enough to feel satiated, then his requirement to recite Birkat Ha'mazon applies only on the level of "Mi'de'rabbanan" – just like the child. Hence, in such a case, the Shulhan Aruch rules that a child would, in fact, be allowed to recite Birkat Ha'mazon on the adult's behalf. It should be noted that this Halacha applies to both men and women. Different views exist as to whether women are obligated to recite Birkat Ha'mazon on the level of Torah obligation or on the level of Rabbinic enactment. Since some authorities maintain that the Torah obligation applies to both men and women, we would not allow a child to recite Birkat Ha'mazon on a woman's behalf unless she had eaten less than the amount of "Kede Sevi'a."
The Mishna Berura (commentary to the Shulhan Aruch by Rabbi Yisrael Kagan, 1839-1933) cites (186:7) different views regarding a case where both the adult and child partook of less than "Kede Sevi'a." In such a case, the father's obligation indeed applies only on the level of Rabbinic enactment, but the child's obligation is one level below the adult's, in that his obligation generally is Rabbinic in nature, and he has also eaten less than the amount requiring Birkat Ha'mazon according to Torah law. The Mishna Berura rules that one should act stringently in this case. Nevertheless, as Rabbi Moshe Halevi (Israel, 1961-2001) rules in his work Birkat Hashem (vol. 2, p. 317), after the fact, if the child did recite Birkat Ha'mazon for the adult in such a case, the adult has fulfilled his obligation and is not required to recite Birkat Ha'mazon after the child's recitation.
Summary: Once a child reaches the age of six his/her parents must ensure that he/she recites Birkat Ha'mazon after eating bread. Generally speaking, an adult man or woman cannot fulfill his/her obligation of Birkat Ha'mazon by listening to its recitation from a child who has yet to reach the age of Bar Misva or Bat Misva. The exception to this rule is a case where the adult partook of less than Kede Sevi'a – meaning, he did not eat to the point of satiation – but the child did eat to the point of satiation. However, if both the adult and child did not eat to the point of satiation, then although preferably the child should not recite Birkat Ha'mazon on the adult's behalf, if he did recite it for the adult, the adult has fulfilled his obligation.