Google defines familiar as:
1. Well known from long or close association
2. In close friendship; intimate
Commonly familiars are animals, though I have heard of instances where friends who are close on more than one level becoming familiars. An instance of a familiar is having a cat from the time you were born, but you are older than the cat. That cat has watched you grow and that cat has been a part of your whole life. That cat is your familiar. He is your friend. If you were to pick up magic, you would call your cat your familiar but otherwise he'd be a cat that you love dearly and are close to.
Familiars are good for protection (not just physically), directing will, and even storing energy if you were to put a collar with an amulet of your making. Animals are able to see through the veil with much greater ease than us, so they are also handy for picking up on things we can't.
Usually the animal picks you. You can't make them love you and respect you. But if you love them very much and if they love you very much as well, then it may be easier than you think for them to become your familiar.
I've got some experiences , but I don't think I'm prepared to share these ones publically, however if you wish to message me to further this discussion, I welcome you to.