Being
A Being was one of the three classifications used by the British Ministry of Magic to catalogue the various magical Creatures that inhabited the wizarding world (the others being Beast and Spirit). The definition of a Being was "any creature that has sufficient intelligence to understand the laws of the magical community and to bear part of the responsibility in shaping those laws". This definition was laid down by Minister for Magic Grogan Stump in 1811, and the Being Division was formed as part of the Department for the Regulation and Control of Magical Creatures.
The definition of "Being" had a long and troubled history. In the fourteenth century, Burdock Muldoon, Chief of the Wizards' Council, attempted to define Beings as any member of the magical community that walked on two legs. Muldoon invited all Beings to a summit to discuss new magical laws, and saw that his definition was unworkable. Goblins had brought with them every creature they could find with two legs, including Diricawls, Augureys, Fwoopers, Pixies, Fairies, and Trolls, and the summit descended into chaos when the Trolls started to destroy the chamber with their clubs.
Muldoon's successor, Elfrida Clagg, attempted to redefine Beings as those that could speak the Human tongue. However, there were still problems at the next summit. Trolls were still classed as Beings since they could be taught the Human tongue, and Jarveys were in attendance. The Centaurs refused to take part since the Merpeople were excluded, as Mermish cannot be understood above water. Ghosts, who had not been classed as Beings under Muldoon since they glided, left in disgust at the Council's emphasis on matters of the living over matters of the dead.
In 1811, Grogan Stump laid down the new definition of a Being, to the agreement of the magical community, along with the definition of a "Beast". However, the Ghosts were still sensitive to being classed as Beings since they were dead, and a new definition, "Spirit" was created for them.
Examples
Humans (Muggles and wizards and witches) were beings, though there were extremists who campaigned for the classification of Muggles as Beasts. Goblins, vampires, hags, giants, house-elves, veela, and werewolves in their human form were also given this status. It is important to note that when there was a full moon, the werewolf in its transformed state would be classified as a beast and not a being.
Merpeople and centaurs were offered this status, but refused because they did not want to be grouped with creatures such as hags and vampires, and were by default classified as Beasts.
Other creatures fit the qualifications that define Being (intelligent speech), such as Acromantulas, Sphinxes, Manticores, and Erklings, but were not even offered Being status due to their extremely violent and even lethal tendencies, and thus they were also classified as Beasts.
Dementors and Boggarts, on the other hand, were neither a Being nor a Beast, as they were non-beings.
Creatures currently specified as Beings
Humans
Muggles
Squibs
Wizards
Giants
Half-giants
Goblins
Dark Goblins
Pukwudgies
European Goblins
Half-goblins
Vampires
Half-vampires
Hags
House elves
Yumbos
Veelas
Half-Veelas
Werewolves - only when in human form; when in wolf form they were regarded as Beasts.
Merpeople - offered the status but refused
Sirens
Selkies
Merrows
Centaurs - offered the status but refused
Behind the scenes
The final definition seems to determine that any creature with sapience can be considered a Being. This concept is noted in various science fiction universes, especially in Star Wars. However, some of the less civilised sapient species do not possess the status of beings, like Banshees or Manticores.
In the German translation, the Being status is translated as Zauberwesen ("magical creatures"). This translation is problematic in-universe, as this classification also includes giants and even Muggles, who have no magical abilities at all. The other classifications are translated Tierwesen ("animal creatures") and Geisterwesen ("spirit creatures").