Magical Paintings or portraits are normal paintings, except that the people portrayed in them move and interact within the painted scene, with other adjacent paintings, and with the outside world.
Characters in the paintings can move from one frame to another adjacent one; at several points, a character is seen flitting from painting to painting to spread news, and in Harry Potter and the Prisoner of Azkaban, one character, Sir Cadogan, travels from painting to painting to show Harry and Ron the way to class.
The characters in several paintings are mentioned by name and are characters in their own right: The Fat Lady who guards the Gryffindor common room, Phineas Nigellus, and Sir Cadogan, are the most prominent among them.
It is also apparently possible for the subject of a painting to move to any other painting in which he or she is portrayed. While it is certain that the portraits that hang in the Headmasters’ study at Hogwarts have this ability, we never learn whether other portraits are able to do this. In some cases, notably that of Phineas Nigellus, there is one inhabitant for, in this case, two pictures. This means that there is always one ’empty’ painting. While it might seem odd to Muggles to have a painting without an inhabitant, this is apparently fairly common in the Wizarding world; Ron remarks, when Harry comments about his Famous Wizards card being blank, “Well, you can’t expect him to hang about all day.”
The two paintings can be at any distance apart, but the inhabitants will still be able to switch. In several cases, people depicted in paintings in Professor Dumbledore’s office travel instantaneously to their images in London. This switching can be useful to carry messages between the locations of the different pictures, or report on what is going on in one of the other places.
Apparently, the inhabitants can travel from portrait to portrait within a given Wizard building; we see this both at Hogwarts and at St. Mungo’s. Note that the portraits in the Headmaster’s office cannot travel from frame to frame once they have left Hogwarts; in Harry Potter and the Deathly Hallows, the portrait of Phineas Nigellus does say that the portrait of Dumbledore cannot travel to the canvas that Phineas is using to visit them. We are led to believe that the portrait of Phineas Nigellus can travel from portrait to portrait within his erstwhile home at Grimmauld Place, but we never actually see him do so.
The limitation on where portraits can travel is clearly necessary, as without these limits, any subject of any painting could visit any other painting. This would provide, in this story, something of a deus ex machina, as Harry would be able to summon and interrogate a portrait of Dumbledore no matter where he was. Dumbledore’s absence is vital for Harry’s maturation.
One must wonder about the case of a subject who has been painted multiple times, such as, we presume, Albus Dumbledore. If there are twenty portraits of him scattered about the Wizarding world, does he appear in each of them in turn? Or are there multiple Dumbledores spread out among the portraits? And would it be possible for them to collide, such that you would have, perhaps, two or three different-aged Dumbledores appear in your portrait occasionally?