As the TV Tropes article so aptly puts it, a woobie "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." It's the character (or maybe characters) that the reader or audience can't help but empathize with. Excellent canon examples of this include Charlie Brown, WALL-E, Remus Lupin, and the majority of the characters of Torchwood.
However, while the woobie is an established character role (or maybe sub-role), turning a character into a woobie when they would not be otherwise (known as woobifying) is a viable charge, unless it is pulled off very well (which almost never happens).
Woobification is strongly related to hurt/comfort. The woobie is given an angsty past or present or something else to cry about, and Mary Sue or his Twu Wuv comes to talk to him and kiss away the booboo. In some often occurring cases, a murderer or sociopath, such as Sweeney Todd or the Joker, is found to be angsting about their behaviour. They are not inherently woobies, but have indeed been woobified. Other examples of this can include Severus Snape and Sauron.
The most frequently woobified character is Legolas. Draco Malfoy, Lucius Malfoy, Harry Potter, and Voldemort tie for a close second.