A furry is someone who adopts an anthropomorphic persona, usually for art, character, identity, roleplay, or some other purpose.
For those who don't know what anthropomorphism is:
"the attribution of human characteristics or behavior to a god, animal, or object"
When dealing with furries, however, we are of course, referring to the animal option there. A furry is someone who identifies with an animal, for instance, as a cat person, so adopts cat like characteristics in their behaviour.
There are a lot of instance of furry in media. For instance, one might argue that the characters of the old children's TV show, Arthur, were furry. Another recent example is the Steam RPG, Dust: An Elysian Tale (as a side note, the flying creature in the top left, Fidget, would not necessarily be considered a furry character because she isn't all that human-like (although she is adorable)- but she's not the point; every other character in the game down to the NPCs is furry)
As a side note, the usual cat-girl paradigm from anime stuff is not typically considered 'furry'. For instance, Katt from the Breath of Fire RPG series is an example of a character who crosses over from normal nekomimi into furry territory, because as you can see, she actually has fur aside from ears and a tail- however Ahri, from League of Legends, would probably not be considered furry because she only has ears/tail as her animalistic traits
Anyway, furries tend to get a lot of undeserved hate on the internet because a few years ago someone, probably from 4chan, popularized the notion that furriness was strictly a fetishistic thing (possibly crossing over into bestiality) and a lot of people got riled up against it without really understanding what it was. There is nothing inherently sexual about being a furry, and they are not actually sexually attracted to animals. So let's just be clear about that before anyone gets too concerned.