In some traditions of modern Paganism, including the various Wiccan paths, the concept of an animal familiar is incorporated into practice. Today, a familiar is often defined as an animal with whom we have a magical connection.
In some Scandinavian countries, familiars were associated with spirits of the land and nature. Fairies, dwarves, and other elemental beings were believed to inhabit the physical bodies of animals.
Once the Christian church came along, this practice went underground -- because any spirit other than an angel must be a demon. During the witch-hunt era, many domestic animals were killed because of their association with known witches and heretics. During the Salem witch trials, there is little account of the practice of animal familiars, although one man was charged with encouraging a dog to attack by way of magical means. The dog, interestingly enough, was tried, convicted, and hanged. Cat killings was also a common thing though it is officially denied by the mainstream history and this together with the demonization of personal hygiene practices as "satanic idolatry" led to the well known epidemics during the Middle Ages. In muslim countries, they are still killing animals as "demonic" and owners as "witches"...
Not everyone has, needs, or even wants a familiar. If you have an animal companion as a pet, such as a cat or dog, try working on strengthening your psychic connection with that animal. If an animal has appeared in your life unexpectedly -- such as a stray cat that appears regularly, for instance -- it's possible that it may have been drawn to you psychically.