The Shulchan Aruch (282:3) rules explicitly that a child may be called for one of the seven Aliyot on Shabbat, provided that he understands the concept of reciting a Beracha to G-d. Accordingly, Chacham Ovadia Yosef rules in "Halichot Olam" (vol. 3, page 120) that a child who is at least six years of age and understands the concept of reciting a Beracha may be given one of the seven Aliyot on Shabbat. He adds that the child may even read the section of the Torah for which he is called. A child should not, however, be allowed to read the entire Torah portion, unless nobody else in the synagogue is qualified. But as for reading the section for which he is called, this is permitted even if other qualified readers are present. Furthermore, a congregation should not call up minors for a majority of the Aliyot; children may called for only as many as three of the seven Aliyot on Shabbat morning.
Our practice is generally not to call minors for an Aliya before their Bar-Mitzva. However, we do rely on this Halacha in situations where a family holds a Bar-Mitzva celebration in the synagogue a bit early, before the boy's thirteenth birthday. In such a situation, we may certainly be lenient and allow the boy to receive an Aliya and read his section, and this is, indeed, our practice.
Summary: A child may receive one of the seven Aliyot on Shabbat morning, provided that he is at least six years old and understands the concept of reciting a Beracha. However, no more than three of the seven Aliyot should be given to children. A child may read his section, as well, but he should not read the entire Torah portion if another qualified reader is present.
