TVTropes definition: "A woobie (named for a child's security blanket) is that character you want to give a big hug, wrap in a blanket and feed soup to when he or she suffers so very beautifully. Woobification of a character is a curious, audience-driven phenomenon, divorced almost entirely from the character's canonical morality.
The Woobie's appeal lies in how it allows the audience to experience catharsis. The Greek philosopher Aristotle proposed that tragedy is popular because it allows people to experience and let out their negative emotions, "cleansing" themselves. The Woobie is popular for this same reason. A story with The Woobie allows the audience to vicariously experience relief from some pain by fantasizing about relieving The Woobie's pain. (No, not that way! Well, okay, sometimes.)"
There always seems to be a line to be drawn with Woobie-dom, between "you poor thing" and "okay, this is getting creepy", aka the difference between Chiyo-chan and Sephiroth. Usually the line is crossed when a fan takes a character with little to no woobie-ish traits and piles them on either as a clumsy way of fleshing them out or because they love the character SO much (and if they're a villain, need to justify their intense love).
What makes a character a woobie to you? Do you gravitate towards seeing males or females as your personal woobies? Where do you draw the line at giving a villain or badass woobie traits to make them more human?