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). Woobifying happens a lot in bad slash, but also in Sue stories where the Sue is a "strong woman" and needs to prove how tough she is by making the men look weak and stupid.
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. Becoming a woobie is in blatant contradiction of their canon characterization. Other examples of this include Severus Snape and Sauron.
Note that it is not a charge to simply have bad things happen to a character, even if those bad things are meant to have the reader feel sympathetic. Woobifying the character means not just trauma, but trauma that drives a character OOC—usually by making him a great deal weaker, more despondent, or generally less capable than his canon self.
The most frequently woobified character is Legolas. Draco Malfoy, Lucius Malfoy, Harry Potter, and Voldemort tie for a close second. (Though to be fair, practically half of the cast from Harry Potter are woobies to begin with.)