Elsewhere is an ambiguous term used to keep citizens in the dark about what happens when someone is "released" from the community. Both Elsewhere and "release" are coverup words. They cover up the truth about the murder of innocent children and the elderly, which is done all in the name of population control. Elsewhere is also used to cover up where people go after they are executed for breaking the rules. For example, when a pilot breaks the rules and flies over the community in chapter 1, the following announcement is made over the loudspeakers: "Needless to say, he will be released" (2). It is automatically assumed that when someone is "released," he or she also goes to Elsewhere.

Since the purpose of the community is for everyone to live a trouble-free life, the way these two words are used with particular phrasing creates a safe way for the delicate ears of citizens to hear and absorb what happens when people leave the community. Incidentally, if people truly understood that Elsewhere means "nowhere" and release means "death," then they would not feel safe. 

Another way the term Elsewhere is used refers to the physical boundaries of the community. Since the community's goal is to exist in its own figurative bubble, it does not teach its children about the world as a whole. However, Lily does meet children at school who come from a similar community, so it can be inferred that other people are living under similar circumstances, but these other communities are not included in the term Elsewhere. Therefore, Elsewhere is any place unknown to the community's understanding, whether that refers to physical boundaries or the place people go after death. People only understand that Elsewhere means somewhere permanent. For example, when Jonas thinks about Gabriel's placement in the community, he explains the following about Elsewhere:

"At least, Jonas thought, after Gabriel was placed next year, they would still see him often because he would be part of the community. If he were released, they would not see him again. Ever. Those who were released—even as newchildren—were sent Elsewhere and never returned to the community" (42-43).

Therefore, Elsewhere is an idea of permanence, whether it is physical or not. Elsewhere means that a person is never seen or heard from again. Elsewhere is significant to Jonas at the end of the book because he must cross the physical boundaries of the community in order to execute the plan he and the Giver come up with. The Giver explains as follows:

"So if you escape, once you are gone—and, Jonas, you know that you can never return . . . If you get away, if you get beyond, if you get to Elsewhere, it will mean that the community has to bear the burden themselves, of the memories you had been holding for them" (155-156).

In the above passage, the Giver uses Elsewhere as a term for the community's boundaries because it is a word that Jonas understands both physically and mentally. Not only must Jonas cross the borders beyond the community's reach to release the memories, but he also understands that Elsewhere means that he's never coming back.

Elsewhere is a universal term for anything that is not the community.  It applies to death, but also to areas beyond the community’s borders.   

When a person is Released, the person is said to go to Elsewhere.

Those who were released—even as newchildren—were sent Elsewhere and never returned to the community. (ch 6, p. 43)

To apply for Release is basically to commit suicide.  Some do it, such as The Giver’s daughter Rosemary.  Essentially, Elsewhere means death.  If you believe that Jonas and Gabe made it to Elsewhere at the end of the book, this means they did die.

Another interpretation of Elsewhere that seems to be supported by sequels is that Elsewhere is a real place, and Jonas and Gabe both survived to go to this other place that is outside the jurisdiction of the communities.

Lowry, Lois (1993-04-26). The Giver (Newbery Medal Book). Houghton Mifflin Harcourt. Kindle Edition.