When you reconsider something, you think about it again or in a new way. You might decline your grandfather's invitation to the circus, but then reconsider when you see that Gunther the famous lion tamer is showing up.
When you reconsider, you usually end up changing your mind about your initial decision or opinion. Your neighbor might be planning to vote for a candidate you dislike; however, if you do a good enough job arguing in favor of a different candidate, she might reconsider her vote. The Latin root of consider is considerare, "look at closely," or "observe the stars." Reconsider adds the "again" prefix re-.