Animals are one of the six realms of existence.
You can see a visual depiction of this in the Wheel of Life:
In this image, Humans are on the top left of the wheel, with animals directly below us.
Also, on top are the Devas (god-like and heavenly realms),
and to the right of them are the Titans, (beings born with power but jealous of those in the Deva realms and always seeking to do war and strife with them)
Then below them, are the Hungry Ghosts or Asura realms, (disembodied spirits, ect)
Follow by below, Hell realms, or those who live in constant agony and severe pain and anguish (I tend to think of Rawandan Genocide here, or the Prison Camps in North Korea)
And then to the left again to Animals.
The important thing to note, is that none of these realms is considered permanent.
Beings are not sentenced to live forever and ever in a hell realm for instance, and even those with a great deal of merit will not stay in "heavenly" realms forever if they do not increase their merit.
It also shows that with the human realm, is considered the best for training, as those in the human realm are the most willing to listen and hear and practice the Dharma as we are not too busy distracted by pleasure such as in heavenly realms or intense pain such as in hell realms.
It is also important that each realm has it's own Buddha, and Buddhisatva's and it is said that the kindest Kanzeon (Boddhisatva of Compassion) is found in Hell, or hell realms.
That is to say that there is always some compassion to be found or something or those pointing the way.
In the center is the pig, snake and rooster, symbolizing Greed, Anger, and Delusion, respectively, which is what keeps the wheel turning.
Becoming free from Greed, Anger, and Delusion, is what allows one to "escape" the constant wheel, and realize Enlightenment, and then become a Buddha.
Around the edge are the various kinds of acts that we get up to in our lives, that keep the wheel turning.
Representing Dependant Origination. These are shown as links or somewhat chain-like
The inner circle next to the center shows that by creating good or bad karma we climb out of lower realms or fall out of higher ones into lower realms, but that this really is just a circle that keeps spinning, and not, in itself a solution to suffering.
In one of the ceremonies in Zen, we offer merit to all the evil and wicked in heaven.
That is to say that even these states are not ideal, and rather temporary planes of existence, and have suffering and Greed, Anger, or Delusion, also.
There is a specific symbolism regarding the demon holding it, though offhand I forget what that is. I'll post later if I remember.
There is also shown that there are people in the human realm leaving the wheel to Buddhahood, as describes what I said earlier.
There is also usually a moon, or Buddha or Bodhisattva in one of the upper corners that shows throughout all this, it is not separate from the Eternal, and that there are Buddhas who point the way.
This diagram is said to have been come up with by The Buddha Himself.
In Gassho,
Sara H