What problem/conflict is the main character dealing with in The Giver?
Another possible answer is that the novel deals with being the same vs. being different. There's no doubt that Jonas is different from everyone else, and he is treated special, and different, and he's uncomfortable with that; lying is the thing he has the most problem with, but when he finds out that everyone lies, it devastates him. The fact that Jonas, or anyone, is different, causes problems with the society; Gabriel is nearly killed as a result of not fitting into the sameness.
It's important to consider, though, some of the things that this society is no doubt attempting to do by making everyone the same. People are to be healthy, intelligent, and everyone has a role in this society. The biggest problem with the society in The Giver, of course, is what happens to you if you're different, and most people would agree that it's not OK what this society does.
Generally, the conflict is the individual vs. society. Specifically, Jonas is struggling against his community that has made everyone the same, taking away individuality. The communitiy has also taken away choice, emotions, and emotional relationships. Gabriel is scheduled to be "Released" because he isn't progressing as he should for his age. This is the event that causes Jonas to take Gabriel and leave the community.
On the other side of the conflict, the people have a community where there's no war, no conflict, and no problems. Everyone is given a role at the age of twelve to become a productive member of the community. The community runs smoothly as long as there is no outside interference, such as the plane that flies overhead.