This is more of an addendum to Mason's answer, but is too long to fit in a comment.
I agree that the logical interpretation of "everlasting" is usually "something which persists forever", but that isn't always the case.
For example, we general consider the terms "eternal" and "endless" to be synonymous with "everlasting". The Doctrine and Covenants (part of the LDS canon) gives an alternate interpretation of these words that doesn't mean "unending" (D&C 19:10-12):
10 For, behold, the mystery of godliness, how great is it! For, behold, I am endless, and the punishment which is given from my hand is endless punishment, for Endless is my name. Wherefore—
11 Eternal punishment is God’s punishment.
12 Endless punishment is God’s punishment.
From this we can see that "Eternal" and "Endless" are variations of God's name. It's reasonable to extend this to include "Everlasting" as well.
With this in mind an "everlasting covenant" doesn't necessarily mean "a covenant which lasts forever", but rather "a covenant that is made with God".
The question of "why did Mormons stop practicing polygamy" is definitely an interesting one, but I just wanted to point out that an "everlasting covenant" can mean something other than "a covenant which never ends".