In this book, the previous Receiver of Memory has to give all of his memories to Jonas. He is also supposed to teach Jonas how to deal with having all these new memories and how to use them to help the community. Jonas asks the previous Receiver what he should call him. The previous Receiver decides that his job should now be described as the Giver of Memory. This is where the title of the book comes from.
Much of the second part of the book is devoted to "watching" as the Giver gives Jonas memories and these memories change Jonas's way of thinking.
His previous job was to recieve memories from the previous Giver.
The previous Giver will transfer all his memories to him, and now the memories are eventually being passed down to Jonas, changing the way he thinks.