A therian is someone who feels a connection with an animal. The connection the person feels with the animal can range from feeling they have been born into the wrong body (similar to those with gender issues) to simply a belief that they were an animal in a past life and still retain some memories or behaviors of said animal.
Therians do not physically become the animal they have a connection with, although some do describe "mental shifts" where they feel their whole thought process becomes that of the animals.
The generally accepted definition is a person who feels or believes that they are a non-human animal, although not physically. The connection can be spiritual or logical.