A Beast is one of the three classifications used by the Ministry of Magic to catalogue the various magical creatures that inhabit the wizarding world. Loosely defined, a Beast is a magical creature that does not have sufficient intelligence to understand the laws of the magical community nor bear part of the responsibility in shaping those laws. The current definition was laid down by Minister for Magic Grogan Stump in 1811, and the Beast Division was formed as part of the Department for the Regulation and Control of Magical Creatures. There are even extremists who campaign for Muggles to be classified as Beasts.
However, there are exceptions. Centaurs and Merpeople both requested to be classified as Beasts, refusing "Being" status because they objected to some of the other creatures they would have to share "Being" status with, such as Hags and Vampires.
Acromantulas, Manticores (formerly) and the Sphinx would have all qualified for Being status since they were capable of intelligent speech, but since they try to eat any person that goes near them, are vicious and bloodthirsty, and become dangerous when the things they are guarding are threatened or when they are given the wrong answer to their puzzles or riddles, they were classified as Beasts. Additionally, werewolves are only classified as Beasts in their transformed state. When there is no full moon they retain their Being status like any other human. It is possible that dementors are given, or even qualified, this status, because of their inhumane behavior and of their tendencies to suck out their victim's souls, but it is unknown if they were classified as a Beast.