In its most general and widespread sense, a furry is someone who actively enjoys works of fiction centered around animals, usually with varying degrees of anthropomorphism, like Watership Down, Secret of NIMH, Disney's Robin Hood, Great Mouse Detective, Star Fox, etc., particularly if the presence of anthropomorphic animals itself is part of the reason they like it. The largest subgroup within there that starts going beyond the average is people who also draw their own pictures or write their own stories with similar styles. The two most common stereotypes on the internet, often conflated with each other, and what most people are referring to when they say it, are people who dress up in fursuits and people who have sexual attractions to anthropomorphic animals. Arguably the latter should only really count if you're attracted especially or exclusively to them, or if that's the number one reason you're attracted to them, but people who are just attracted to characters that are attractive anyway that happen to be anthropomorphic animals (e.g., Krystal and Gadget for most people) tend to get grouped in there too. A very small but very vocal subgroup are the "otherkin" -- people who actually believe that they are animal spirits within human bodies, that they were born as the wrong species, that humans are evil and only animals are good and pure and stuff.
Technically, the term "furry" only applies to anthropomorphic cats, dogs, foxes, wolves, etc.; the lizards and dragons are scalies, the birds are avians, and so on, but furry is usually used as a catch-all term for all of them.
TVTropes actually has a pretty thorough and fair overview over
here.
