Euphemism is the use of a polite, indirect expression which replaces words and/or phrases that are considered harsh, impolite, or which suggest something unpleasant. For example, "kick the bucket" is a euphemism commonly used when somebody dies. Generally, a euphemism is not something that should be interpreted literally.
The previous post does a nice job of listing many of the euphemisms found with the story. I'll try to add a new one, and I'll focus on the euphemisms in the story that are specifically used when referring to impolite or unpleasant topics.
"Release" is probably the best example of a euphemism from The Giver. It sounds really pleasant. Fishers that want to fish but not kill the fish practice a technique known as “catch and release.” The fisher will catch the fish, celebrate, and then release it back into the water alive. In Jonas’s society, though, “release” has nothing to do with freeing a person alive. “Release” means death. More specifically, the euphemism refers to death through euthanasia. If somebody dies through natural or unintended consequences, then it is referred to as “loss.” In both cases, a person is dead. While “loss” is a euphemism in the book, it isn’t a blatant lie in my opinion. “Release” is a lie. The person isn’t being sent away to some other location; however, most of the people are led to believe that a released person has been sent to a pleasant “Elsewhere.” Perhaps that is indeed true, but that would definitely depend on whether or not the reader believes in some kind of afterlife.
A commonly overlooked euphemism from the book is “transgression.” A transgression is some kind of violation of a rule or a law. It could be something serious like murder, but a transgression could be a small violation too, like a student blurting out an answer without raising his/her hand. If you look at some synonyms of the word, it seems that “transgression” usually doesn’t connote a serious offense. “Impropriety,” “infraction,” and “misdeed” are all synonyms of “transgression.” The word is only used three times in the entire text; however, based on the first usage of the word, it’s obvious that “transgression” in The Giver is not a small offense.
“You know that there’s not third chance. The rules say that if there is a third transgression, he simply has to be released.”
That tells readers that a person’s “transgressions” are punishable by death. The other uses of “transgression” in the story refer to small things like not wearing shoes on the proper feet. But based on that first use, it’s clear that the word is being used for something more serious, like a “crime.”
Another euphemism in this story is “Stirrings.” This euphemism is replacing something like “sexual urges,” and “puberty.” Granted, that is an awkward topic, but in The Giver, that natural change is so taboo that the culture suppresses the feelings with drugs and suppresses the topic through the euphemism.