1. The idea that the familiar is a vessel for a part of the soul of the witch (no prizes for guessing where the inspiration for JK Rowling's horcruxes came from). The link between the witch and herself is strengthened by the bond and the familiar is used to do the bidding of the witch. I don't do it personally but I know witches who claim to use their familiars as their "eyes" and send them into the world to send back what they see via a telepathic link. This idea of storing part of the self is also used in inanimate objects and some witches will give life to a soul doll which may also be put to use to serve specific needs of the witch. I personally find this a bit far from my own personal practise to be comfortable - I don't have a strong enough link with any animal as to actually be able to see through its eyes. But this definition works for some and may work for you.
2. There is also the idea that the familiar is a guide from a past or future life that is drawn toward working with the witch and who seeks her out to offer support and guidance. A variation on this is that the familiar is a reincarnated soul mate of the witch who happens to be an animal in this life but who wishes to retain the relationship.
3. A more practical viewpoint is that a familiar is any animal employed in magical usage with his/her owner. So a cat who falls asleep on your lap is a furbaby and a cat who gets involved in ritual is a familiar.
4. The familiar is often confused with the idea of the fetch which is best described as a entity separate in spirit to the witch but linked to her (for want of a better word) soul by a connection between them. The fetch can manifest itself as an animal but this is likely to be a conscious choice to deepen the connection and not through any affinity with the chosen animal itself. A fetch does not have a corporeal body though it may borrow one if required.The spirit of a familiar will be resident in the allotted body for the length of the body's life span in much the same way a human soul is resident inside a body until the body ceases to live.
5. Some witches believe their familiar not to possess a physical body and consider an animal spirit to be their familiar. This explanation of the term blurs with the concepts of the animal guide or the totem and can cause confusion. I have to say it contradicts my own view of a familiar - to me a familiar by definition has a physical body. Some witches consider a demon to be their familiar -though this is very rare as demons do not play a large part in Trad Craft.
So that gives you an idea of different ways the term familiar can be defined. Some witches take the term even more broadly to encompass any animal loved by the witch but it is more common to believe the familiar is a spirit or a soulmate housed in the body of an animal for the duration of its relationship with the witch in this incarnation.