(I am not a furry, but I’ll try my best to explain what I know.)
People who really like anthropomorphic animals (animals with human features).
Generally what I’ve heard from others is “If you want to be a furry, you’re a furry.”, so the following are not requirements.
Most have a “fursona”. It is an animal character that sort of represents you. Like your own personal mascot.
It may be whatever color and specie you like. Even a hybrid between species. A person can have multiple fursonas.
Generally furries like to draw these characters, or even some more generic ones like for example Nick Wilde from Zootopia.
Some furry artists accept drawing commissions, i.e.: you can pay someone to draw your character.
Some even have fursuits. That is an expensive suit of your fursona, so you can fully be your own private mascot. By expensive, I mean multiple thousands (yes, freedom currency). Of course, it’s made by people, and you are people. DIY is also an option, and if you get skillful enough… money. Though it doesn’t have to be a fullsuit. It could be head-only or partial (head, paws, feetpaws, tail). Also digitigrade suits are usually more expensive than plantigrade, but let’s not get too far.
Not sure if I gave any good explanation, but certainly better than that episode of CSI xD.