Number One was a male Pit bull dog and Jean-Luc Picard's pet in 2399. During Picard's retirement, Number One lived at Château Picard, sleeping on a mat in Picard's bedroom and accompanying him in the vineyard. Picard used French when speaking to the dog.
Although Picard called him an "old dog", Number One was still prone to capturing small mammals and bringing them into the house. (PIC: "Remembrance")
As Picard headed off once more on a new mission into space, he left Number One behind at Château Picard to be cared for by Laris and Zhaban. (PIC: "The End is the Beginning")
Background information
Patrick Stewart, an advocate for pit bulls, suggested that Picard should have a dog on the vineyard, and that it should be a pit bull. [1]
Writer/producer Michael Chabon said that the decision to name Picard's dog "Number One" came in the first draft of "Remembrance". (TRR: "Remembrance")
Dinero, who plays Number One, is a rescue dog, with little previous film experience. "Remembrance" director Hanelle M. Culpepper described DeNiro as "very green" (that is, inexperienced) and "a challenge sometimes, but super, super sweet." Michael Chabon agreed, saying, "He was the sweetest dog. He was not a great actor." Culpepper noted that Number One was present in more scenes in the original script than in the episode as filmed, Culpepper noting, "We both agreed, let's leave the dog out of this one." (TRR: "Remembrance")