An anthropomorphic personification personifies a phenomenon or a concept. Over the worldly happenings of the named phenomenon an anthropomorphic personification may have far more power than the gods. For example, Death appeared when the first mortal creature came into existence in the universe. Death is far older than any of the gods (who come into existence by belief of and worship by sentient creatures), and in Death's house there are lifetimers for the gods just as there are lifetimers for people. An anthropomorphic personification may work on the very fabric of reality so that the phenomenon happens; for example, Time destroys and re-creates the universe moment by moment, so that every moment the universe is new, and the moments form into the phenomenon that is perceived as time. An anthropomorphic personification may also preside over events; for example, War goes and watches major battles, and Death cuts free the souls of those who have died but who cannot leave their bodies.
Anthropomorphic personifications are also connected to the flow of the disc itself and subsequently can sense something not right in the world when humans and magical beings wouldn't notice. A major example of this is in Hogfather where Death is able to hear a "discord in the song of the universe" even when he is at the bottom of the ocean. They are also well versed in magic and the laws of magic, even surpassing Wizards at times. Indeed wizards will sometimes summon Death (to his annoyance) to consult him on different things.
There is also a long-standing animosity between Anthropomorphic personifications and the Auditors of Reality although most of this appears in the form of battles between Death and the Auditors. The anthropomorphs are not fond of the Auditors because they break the rules and have tried on more than one occasion to "assassinate" specific Anthropomorphic personifications such as Death and the Hogfather. Anthropomorphic personifications do, however, get on well with one another and have a mutual respect for each other and what they do. There is also a strong sense of loyalty in Anthropomorphic personification circles and they will go to great lengths to help each other even outside their own fields, such has when Death assumed the Hogfather's duty to ensure the sun would rise and to make sure the Hogfather would continue his duties.
Another reason Anthropomorphic personifications are despised by the Auditors is because they have a capacity to evolve and grow as individuals. It was this very reason in Reaper Man that they tried to replace Death because he had evolved too much in personality and they thought it didn't befit his job. Fortunately, Death was able to keep his job thanks to appealing to the Death of Deaths that caring for the job made it work properly. Azrael agreed and overruled the Auditors. Other Anthropomorphic personifications have evolved from what they originally were; the Tooth Fairy started out as the anthropomorphic personifications of fear or the first The Bogeyman but after so long decided to protect children by guarding their teeth from the possibility of controlling magic. This indicates that as an anthropomorphic personifications goes through life it will develop its own unique personality or as Susan put it in the Hogfather, they stop being what people want them to be and start being what they want to be.
All Anthropomorphic personifications have their own specific realm or kingdom where they spend their time when not on "active duty" where they can rest and monitor things in their dominion without needing to be there in person. As with an Anthropomorphic personification's physical form the shape of the realm is affected by belief. For example, Death's home is dark, Gothic and gloomy because thats how people see his realm; just has the Tooth Fairy's home is based on children's beliefs like blue skies and orange fish. These belief however in certain cases mean that certain Anthropomorphic personifications can not travel to each other's realms due to conflicting beliefs. For example the Wintersmith could not visit the home of the Summer Lady because her home is a place of heat and fire, and Death can not visit the Tooth Fairy's home because children do not understand the concept of Death so there is no Death there. Also spending too long in a Anthropomorphic personification's realm will have side effects on a normal person, such has when the robbers in Hogfather became childish because of being in the Tooth Fairy's kingdom too long. An anthropomorphic personification also has awareness of those within their realm as the Tooth Fairy became aware of intruders when they disturbed his sleep and picked them off one by one and Death knew Rincewind and Twoflower were in his home.