This is the notion that some people are saved and others are not; all according to God's choice.  God elected, or chose, those who would spend eternity with Him.  The nature of election is widely debated however.

Calvinists claim that man is unconditionally elected not based on any merit.  Arminians claim that God foreknew those who would follow Him and then chose according to that knowledge.

There are others who believe that references to election in the Bible are not concerning individuals at all.  Letters of the New Testament addressed to "the elect" are not addressed to individuals but rather the church as a whole.  God elected the church for salvation in eternity past and did not determine each and every individual who would be saved.  When they are saved, they become part of the elect.