For the elderly and some infants, Elsewhere is symbolic of death. They are taken to the Releasing Room and euthanized (put to death) because they are no longer useful, or a baby isn't progressing the way they feel he/she should (like Gabriel).
Elsewhere is also symbolic of nonconformity. In Jonas' community, sameness is stressed in every aspect of life. Places outside of the community may have different rules or may allow people to make certain choices for themselves.
"Elsewhere" represents an unknown world where those who leave the community go. The old, weak, and troublesome are "released" there - although in reality they are murdered, the people are led to believe that they are simply choosing to leave the community in a rite of passage, traveling to "Elsewhere". In this case, "Elsewhere" is similar to a belief in an afterlife, an explanation as to what happens when a person leaves life as it is known.
"Elsewhere" also represents a physical place outside the community's boundaries. It is the place to which Jonah travels after his escape from the society.