Phineas Nigellus is one of Sirius's ancestors. He was the least popular Headmaster Hogwarts has ever had. Even so, his portrait hangs in the Headmaster's office at Hogwarts. Since there is another portrait of Phineas at Number Twelve, Grimmauld Place, he can go between Hogwarts and Order headquarters to deliver messages. The thing is, Phineas has a much more integrated personality than, say, Mrs. Black's portrait.
He appears to have the ability to reason and also to resist. He obviously doesn't approve of Professor Dumbledore's Gryffindorish policies, so he wastes a lot of time passively resisting Dumbledore's orders. This causes the other portraits in the Headmaster's office to shriek at him, since they all feel that it is the duty of the portraits at Hogwarts to help the current Headmaster. But Phineas isn't convinced. He does admit, after watching Professor Dumbledore curse Professor Umbridge and Fudge and escape Hogwarts, that, "I disagree with Dumbledore on many counts but you cannot deny he's got style" (28.265).
Throughout the whole novel, Phineas has been an obstructive little toad. But then, at the end of Book 5, when he discovers that Sirius has been murdered, he says, "Am I to understand [...] that my great-great-grandson – the last of the Blacks – is dead?" (37.62). He can't believe it, and goes to his portrait at Number Twelve to look for him. Harry imagines Phineas at Number Twelve: "He would walk, perhaps, from portrait to portrait, calling for Sirius throughout the house" (37.66). J.K. Rowling tells us that the portraits are snapshots of people who have died, without the full personality that a ghost would have. But Phineas's portrait seems like more than that: he appears to have the human responses of denial and perhaps even sorrow. This final moment with Phineas humanizes him for us; we feel pity for him. But he's also a portrait, which means that it's hard to know how much real feeling he might have.