Not to be confused with people who have a "yiff" paraphilia.
The broadest definition is an "anthropomorphic enthusiast" the same way a trekie is an "star trek enthusiast".
Some might say they are a furry because they collect stuffed animals or "plushies". Some might say so because they enjoy anthropomorphic films/television programmes such as Balto or The lion king.
Like most fandom cults they hold conventions where they get to meet eachother and discuss anthropomorphic topics, dress up in anthropomorphic costumes aka "fursuits", play dance-dance revolution and have a good time.
Some people collect stamps I guess.
The lion king was the only decent furry movie Ive ever seen.