In English, you sometimes add "pet" in front of an animal name so that people know it's not a wild or feral animal, but there is no direct equivalent in German. If you have a pet cat, in German, you will just say you have a cat, or you will call it "my cat", and it's usually understood as being a pet by the way you talk about it. If it's an animal that people don't usually have as a pet, like a monkey, you may want to specify it though, so you could add "als Haustier" (as a pet) after the animal name: "I have a pet monkey." = "Ich habe einen Affen als Haustier." = "Ich habe einen Affen."