In my tradition, the familiar spirit is one of three things.
The fetch-beast is one, a fairy lover. The fetch is your other self; Basically, it represents you. It is the double of you, an astral and etheric self which can sometimes take the form of an animal, animal-like figure, or person.
It can also leave your body (i.e., spirit flight). The fetch is an important part of hedgewitchery as it is the form we take when we travel to the otherworlds. It is a term witches use to describe their wandering soul, the eternal part of the self.
Isobell Gowdie, a 17th century Scottish witch, was reknown for having a fetch of a hare. To cultivate a relationship with it, you must be able to recognize it and call it, in a intimate and almost sexual relationship, loving yourself. My own fetch manifests as a Mare with a rotted face, or a pale woman with a similar mares head. I can assume this form as I please when hedge crossing.
There is also the Fetch Lover. This is the higher self, a composite of everything your fetch-beast and it's physical counterparts are.
Another familiar is the imp, or a spirit alliance. These spirits may work with you out of nowhere, or may be gifted to you by a higher power (in my tradition, this is the Witchfather, the Folk Devil). They are demons or fairies (or both, as I don't consider them that different) There are many ways to gain one such spirit. I work with an ancestral spirit, a fairy shade, I call grandmother Jaquabit.
Even the gods have familiars, animal spirits and daimons (little devils) that serve them. Interfacing with a physical animal or it's parts, that is holy or representative of a entity, is interacting with one of that gods many fetches.