What genre is The Giver?
First of all, understand that a book can be more than one genre. The boundary lines of genre can be crossed. Novels can have elements of many genres within them.
First and foremost I would say The Giver is a dystopian novel. Generally speaking a dystopian novel has a setting that should by idyllic, an attempt at utopia has been started, but through some lacking element -- usually something like freedom of choice. In the Giver, everyone has limited individual choice, freedom, love, etc. The only person who sees things clearly (as illuminated by the vision in black and white and color) is the keeper of history (which of coarse Jonas becomes.)
It is also children's literature. Children's literature is usually defined as books written for and used by children. While there is no one set of ideals that makes a book children's literature or young adult literature it often has a young protagonist.
Bildungsroman is a specific form of literature where the protagonist goes from innocence to experience (both psychological and moral growth.) When Jonas becomes keeper of cultural history this process begins. As he begins to see the dystopian elements of his society, as his need to escape/change increases so does his knowledge.
Science Fiction -- The Giver has a futuristic society where people are genetically engineered and the climate is controlled. While the novel does not go into the "how" this is achieved it still allows for a science fiction setting. Science fiction is fiction which is often based on hypothetical (potentially possible) science and often has a future or alternative time setting, application of new or theoretical scientific principals, and sometimes involves political or social systems that differ from our own. The Giver clearly fits these elements.