Therianthropy is categorized by a deep integral or personal belief that an individual is in some way and to some degree, a non-human animal. Most therianthropes(AKA therian) are able to 'shift' into their theriotype, which is the animal they identify to, but this is NOT the main thing of therianthropy.
The word Therianthropy originates from a portmanteu of the Greek words Therion and Anthropos (animal and human respectively) which, combined, loosely translate to "human animal" or "animal man". It was originally used to describe creatures or men who could change shape and become something else. Werewolves, werecats, and other werecreatures could be described as Therianthropes, though it is a very broad term and it is more acceptable to use their species specific labels.
Therianthropy stretches back to the dark ages, allegedly being used in werewolf trials of the 16th century, and to refer to shapeshifters throughout history. There are currently no documented cases of the term being used to describe similar situations to that of current, modern day Therians, but there are many well documented examples of what could be considered Therianthropy.
Modern day usage of the word is thought to have originated in the early 1990's, when users of the werewolf newsgroup Alt.horror.werewolves began discussing their spiritual and mental connections with their animals, asserting that they were in fact their therioside. At first, the term Lycanthrope or Werewolf was most commonly used, but Therianthropy became more widely accepted as it catered to all species, and not just wolves. AHWW is credited the birthplace of the online Therian community.
Most therians today are people who belive that in some way, they are an animal. They could be the reincarnation of a dog or a fox. Some think that their soul ended up in the wrong body. They are very real and can often have what are called shifts. This can be a mental shift where they start thinking like their animal, or phantom shifts where they feel their missing limbs like tails or ears. Few therians believe physically shifting is possible, but a vast majority perceive it as impossible.