Borderline Personality Disorder
Its a type of personality disorder. From the DSM-IV TR "A personality disorder is an enduring pattern of inner experience and behavior that deviates markedly from the expectations of an individual's culture, is pervasive and inflexible,...,is stable over time and leads to distress or impairment."
Borderline people are highly unstable in virtually all senses (relationships, self-image, etc) and highly impulsive. They also have abandonment issues, and can be suicidal, and self mutilate. People with Borderline Personality Disorder, often also suffer from other personality disorders.
They are people you really want to avoid if at all possible, but they can be difficult to identify. I've known a few in my life who were definite candidates for it, and made the mistake of dating one though I didn't realize it at the time.