A dandere character is one who is normally quiet and silent, possibly to the point of coming across as an Emotionless Girl at times, but will suddenly become talkative and sweet and cute when alone with the right person if they manage to push the right button to get them to pour their heart out, revealing that she (or he, of course) is actually just shy.
The name comes from an abbreviation of 'danmari deredere'. 'Danmari' means to be calm or to shut one's mouth up all the time, while 'deredere' means lovestruck.