In the mormon belief system, a celestial marriage is a marriage that extends for all eternity, including "the afterlife".
The book of mormon says there are three 'levels' of heaven, the highest one being the celestial level. To get into the celestial level you must (among other things) be in a celestial marriage. The celestial marriage must be performed by "one who holds the sealing powers and authority" and be held in a "proper place" - a mormon holy temple. If these prerequisites are not met the marriage will last only for this life, and thus it will negate the chance of celestial exaltation. Partners in a celestial marriage are sometimes called "sealed".
It is interesting to note that Mormons are required to marry to acheive celestial exaltation:
All Mormon men who desire Godhood are required to marry; if they do not, their leaders have taught that their actions will be displeasing to God. ... Those who choose to remain single or do not enter into the covenant of celestial marriage while on earth are no longer in obedience to God or to LDS authorities. They will not advance to Godhood, but will be given menial tasks as angels for all eternity.1
It is also interesting to note that some celestial marriages are polygynous. Both celestial marriage and polygyny are described in section 132 of the Doctrine and Covenants, but they absolutely do not have to be, and these days they are not. (See Mormons and Polygamy).
1. http://www.mrm.org/articles/celestial-marriage.html
http://www.yrulds.com/home/marriage/definition.htm
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