To “rue” is actually to regret, so “rueful” means regretful.
The first time the word is used is describing Asher, who was smacked whenever he used the same word asking for a snack as a three year old. The incident, which caused him to stop speaking for a time, is brought up at the Ceremony of Twelve.
Asher nodded ruefully, and the audience laughed aloud. Jonas did, too. He remembered, though he had been only a Three at the time himself. (ch 7, p. 54)
Asher regrets the experience, and probably the fact that it has been brought up years later, because he is embarrassed.
The Giver is also rueful when he tries to explain to Jonas about his predecessor, the unsuccessful Receiver of Memory Rosemary.
The Giver chuckled ruefully. "The explanations are difficult. The whole thing is so beyond one's experience. But I tried. And she listened carefully. Her eyes were very luminous, I remember." (ch 18, p. 141)
The Giver has regrets about Rosemary because he was not successful with her. He could not bear to cause her pain, so he did not properly prepare her for the memories. He has to live with her death for the rest of his life.
When Jonas suggests that they are going to have a sharing of feelings ceremony when he stays with The Giver for dinner, as all people in the community are required to do, The Giver has a strange reaction.
The Giver gave a rueful, anguished, empty laugh. "Jonas, you and I are the only ones who have feelings. We've been sharing them now for almost a year." (ch 20, p. 154)
Here, The Giver is regretful because he realizes the irony in the telling of feelings requirement. The feelings shared are minor, and not real feelings, and they are shared only so they can be dismissed.
Finally, the last person described as rueful is Jonas’s father, as he describes Gabe’s return to the Nurturing Center to see if he will sleep for the night.
Father gave a rueful laugh. "That's an understatement. It was a disaster. He cried all night, apparently. The night crew couldn't handle it. They were really frazzled by the time I got to work." (ch 21, p. 164)
Jonas’s father regrets Gabe’s not having made enough progress, but does not really seem to care that the baby he has raised as his own for a year is going to be killed. This is what causes Jonas to flee with Gabe.
Lowry, Lois (1993-04-26). The Giver (Newbery Medal Book). Houghton Mifflin Harcourt. Kindle Edition.