The term "furry" means any anthropomorphized animal. They are also known as "funny animals," but this is an outdated term. Since a furry is an anthropomorphic version of an animal, we can determine what a furry character is by carefully defining our usage of "anthropomorphic
By definition an object is "anthropomorphic" if it has been given human attributes; these usually include the capacity for rational thought, emotions, and an upright posture. An anthropomorphic animal, therefore is a hybrid that is somewhere between a human and the original species.
Examples of furries abound, and these can surely cement the concept for any readers that stopped to re-read the above paragraphs. Most cartoon characters are furries (e.g. Donald Duck, Wile E. Coyote, Pink Panther, Quickdraw McGraw, etc.). Many cartoon characters are quite removed from the actual animals; their charicatured forms still retain many animalistic traits, however, and this identifies them as furries.
Many comic book characters (e.g. Teenage Mutant Ninja Turtles, Omaha the Cat Dancer, Usagi Yojimbo), movie characters (e.g. the Rippers in "Tank Girl", Barf in "Space Balls"), and mascots (e.g. the San Diego Chicken and Smokey the Bear) are also, at a basic level, anthropomorphized animals.
Finally, furries are also commonly found in artwork, literature, and mythology (I won't bother to list examples here; I think you've got the idea by now). These are diverse categories, and this is one reason that it is hard to develop a comprehensive notion of what constitutes a "furry".
I like to draw furry artwork.