A dandere character is one who is normally quiet and silent, possibly to the point of coming across as emotionless 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 they're actually just shy.
A dandere will be more or less a normal person, but, will not talk unless spoken to/required to. They will avoid talking because of their shyness, but if need be, they will speak. Also, a dandere tends to feel more power in numbers. If they're with a group that they're comfortable with, then the shyness factor of said dandere will very likely decrease if another, unfamiliar person is introduced at that time.
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.