The word furry has several meanings, dependent on the context in which it is used. Predominantly, it means “consisting of or resembling fur”. It also pertains to an interest in anthropomorphic animals, mythological or imaginary creatures which possess human or superhuman capabilities.
Someone who says they are furry is generally expressing an interest in anthropomorphic animals and/or creatures, or perhaps some affiliation to furry fandom. They may express that interest in a variety of ways – through art and stories, to roleplay and performance. How deep or meaningful an interest in Furry is varies greatly from person to person.
From within the fandom, anything classified as an anthropomorphic animal and/or creature could be called a furry. That could be anything from Scooby Doo to various sports mascots to Chuck E. Cheese® or Rapid T. Rabbit. Since mainstream characters are generally not created with furries in mind, the furry context is presumed by most outsiders to not be present despite subcultural interest.
Heres a great video about what a furry is.