Familiar Spirit: Any spirit that has been called for you, by your own action and ritual, or another person's ritual work. Familiar spirits are a classic part of most forms of wtichcraft, some hermetic practices, and most forms of old school demonology. It's one reason the Goetia has so many Demons listed as "gives good familiars" because a spirit that is "familiar to you" knows you, works with you, you trust it and it trust you. There is a bit of a bond between the two after time (keeper and spirit). Older practices of gaining a spirit familiar are very hostile, so I don't like the older methods, my methods use often the tools of the old, but in a more gentle, compassionate, and above all else consented way. I don't believe in enslaving spirits, however OLD SCHOOL familiar spirit methods do use that, so its good to let people know where I draw my own lines.
Animal Spirits: These are spirits of animals that once lived, dogs, cats, bears, wolves, eagles, and things of this nature. They may transform onto death, such as the fabled birds of fire, or they may retain their old form. With death they gain a bit of a greater connection to the ALL, as their minds were uncluttered (such as ours are) and it gives them both the insight of their own life, as well as the insight of that less cluttered mind to the all. When asked to come into service they select a person who is best suited for them, or they are a free agent just looking to help SOMEONE (Preconjure animal spirits fit this) so long as they can get the opportunity to. This would be my "Animal Companion Conjures"
Spirit Guides and Guardians: These are spirits that are already with a person acting on their best interests. They are basically your "birth team" or the group you start with regardless, and their involvement is very highly determined by how much they are embraced or denied by the practitioner. Someone who works with them often will find them amazingly powerful allies, someone who doesn't will find they just don't exist.
Personal Animal Spirit: This is the spirit form we take when we move to an animal form. Everyone has a different one, and in that every form is unique. Not only shape shifters (in the planes) do this, but often that form gains a bit of independence, a kind of "second self" with our memories and experiences, will and compassion, and our heart and soul being one, but acting independently. You'll often hear a shaman say "that was my wolf" and when you ask them what its name was, they will say "it's name is my name."
Familiar Animals/Animal Familiars/Familiars: This is a classic term in witchcraft referring to an animal with which you have an amazingly strong bond, and who will put themselves on the line for you. I actually refuse to use this term with my cat because I would rather take the hit for her, but I recognize that by all manner of statements, she still fits the description. I suppose it could be said that a good "Familiar Animal" would be one you WOULD put your life on the line for, because that bond is so strong. They often help with magick (Valkyrie my cat has actually re-arranged an altar once for better power, specifically moving crystals into new places on a larger grid so...) or are drawn to the practitioner during acts of magick.