From the alt.fan.furry FAQ:

WHAT IS A "FURRY"?

"Furry" when used as a noun seems to refer to one of two things:

  1. An animal-like character known as a "furry"
  2. A person who is a "furry fan"
The latter is easy enough (knock on wood) to define: A person who particularly enjoys stories, pictures, dolls, video games or whatever concerning "furry" creatures. Defining a "furry" creature is somewhat harder, though. There are several definitions depending upon which "camp" in furrydom, for lack of a better term, you might be in.

The basic definition for a "furry" is an anthropomorphized animal character. In other words, an animal character given human-like attributes, such as sapience and often a humanoid form. The term "furry" is a misnomer, as a creature does not need to have fur to be "furry" in this sense. Other terms sometimes interchangeable with a "furry" in this sense are "zoomorph", "morph", "anthropomorph" or (debatably) "funny animal".

The core definition of a "furry" seems to include basically humanoid-formed creatures with animal faces, fur/scale/feathers/whatever, and often appropriate tails, wings, claws, etc., able to speak, and with a human-like personality, though quite often with "quirks" hinting at the real-life animal upon which the character is based.

A broader definition will sometimes include other odd creatures that simply have some sort of animal features in their makeup. Such would include mythical creatures such as centaurs, manticores, satyrs or harpies, all of which have human faces though more-or-less animal-like bodies. This broader definition might also include the human-like characters that appear in some Japanese animation that have an animal tail and ears, but otherwise look about as human as any other anime character.

One of the narrower definitions held by some is that in order for a character to be truly considered "furry", the character must exhibit animal-like characteristics in behavior. Optionally, the fact that the character is an "animal" must be a major ingredient to the story. This is exhibited in a frequent criticism of "furry" stories by those who hold this view: Many stories, while featuring characters fitting the core definition of "furry" given earlier are criticized as being "humans in animal suits" if their behavior isn't distinctly animal-like in some way.

What does this have to do with me?

I’m a furry. I feel a strong identification with bears. I eat like them, I sleep like them… I wake up like them… I’m told that sometimes I even look like a bear. I consider the spirit of the bear to be my protector. It’s who I am, and what I do, and I have Wolfie to thank for introducing me to the fandom. He took me to Anthrocon, encouraged my art, made me feel less weird about my affinity with my ursine side.

Furries aren’t freaks or perverts, no more than any other subgroup of the population. They’re just people, like everyone else, with their quirks and fetishes, likes and dislikes, odd tastes and all the differences that make the world what it is.