The bearded elder is The Receiver of Memory, the person who holds all of the community’s memories.
Jonas’s community is not like ours. Everyone exists under the conditions of what they call Sameness, which means that there are tight controls on every aspect of community life and all choices are made for people to ensure that everyone is the same. There is no sense of history. All of the community’s history exists in memories that are held in one person, the Receiver of Memory.
Almost no one in the community knows the Receiver of Memory, because he lives apart from the other community members. His isolation is partly self-imposed, because he alone has feelings and emotions that others can't understand. The rest of the people in the community do not feel in the same way, because they are taught from a young age to control their feelings.
Jonas does not really take notice of the Receiver of Memory until the Ceremony of Twelve, when he is selected to be the next Receiver.
The Committee of Elders was sitting together in a group; and the Chief Elder's eyes were now on one who sat in the midst but seemed oddly separate from them. It was a man Jonas had never noticed before, a bearded man with pale eyes. He was watching Jonas intently. (Ch. 8)
The old man is watching Jonas because he was largely influential in choosing him as the next Receiver. He alone knows what the job entails, and told the elders what they would need to know in order to choose the right person. Jonas was selected partly (or mainly) because he actually was related to the current Receiver. Evidence of this is seen in the pale eyes that they both have. The Receiver also was aware that Jonas had the Capacity to See Beyond, which is rare but needed to harness the memories.
Once Jonas becomes the Receiver of Memory, the old man tells him to call him The Giver. This is because a large part of Jonas’s training involves receiving memories that are given to him by The Giver. The Giver has all of the memories, but needs to begin transferring them to the new Receiver, Jonas.
The Giver is training Jonas to be the Receiver of Memory, one of the community’s most important jobs. The Elders comment about the fact that they are not directly involved in Jonas’s training.
"But the Receiver-in-training cannot be observed, cannot be modified. That is stated quite clearly in the rules. He is to be alone, apart, while he is prepared by the current Receiver for the job which is the most honored in our community." (Ch. 8)
The Receiver of Memory advises the community on problems the elders cannot make decisions about because they have so little information, since there is no history in the community. The Committee of Elders ask the Receiver of Memory’s advice on issues such as changes to community procedures. For example, when they considered expanding the population, he advised them against it to prevent overpopulation and hunger, concepts only he has access to from the past.
The Giver is extremely important to Jonas. In addition to being his mentor and teacher, as Jonas learns more about the real world while gathering memories, The Giver becomes like a father or grandfather to him. The Giver is compassionate and wise, and guides Jonas through the difficult awareness he has to come to with the realization that his community is far from the perfect, orderly world he thought it was.
In time, The Giver becomes the real family to Jonas. Jonas is aware that they are related, and that he is related to Gabriel too. The Giver and Gabriel are the only people that Jonas can be authentic with, because no one in the community has any emotions. Jonas even asks The Giver to come with him when he escapes. He explains why he can’t.
"But don't you want to be with me, Giver?" Jonas asked sadly.
The Giver hugged him. "I love you, Jonas," he said. "But I have another place to go. When my work here is finished, I want to be with my daughter." (Ch. 20)
The Giver is referring to Rosemary, the former Receiver of Memory in training. He explains to Jonas that he does care about him, but that he cannot go with him. However, he has prepared Jonas well, and Jonas is able to escape successfully and save Gabriel and the community. None of this would have been possible without The Giver.
As one of the only people in the community who really knows what is going on, The Receiver of Memory is an important job. When Jonas takes the torch from the old man, he realizes that there is a reason the job is so special. The Receiver of Memory is supposed to advise the community on how to change slowly, but Jonas and The Giver decide that enough is enough. The community needs to learn how to feel and be human again.