12th Century symbolism
Even today, we associate characteristics or virtues to certain animals or objects eg. "pride of a peacock", "diamonds are forever", "as tough as an elephant". Some of these date back to medieval times (when assigning virtues was quite a fad), others are more recent.
A good source to find the symbolism associated with animals (and occasionally gemstones too) is in a bestiary. A good example of a 12th century bestiary on the internet (with excellent pictures and translated text) is the Aberdeen Bestiary.
Many of these are associated with Christian values, as many medieval bestiaries were written as Christian moral lessons. Earlier non-religious symbolisms may have been adopted within these symbolic bestiaries, and others exist, but are harder to research.
The main symbolism of the peacock is pride and vanity, when its tail is displayed.
But the peacock can also represent learned teachers, as "The peacock has hard flesh, resistant to decay, which can only with difficulty be cooked over a fire by a cook, or can scarcely be digested in the stomach, because of the heat of its liver. Such are the minds of teachers; they neither burn with the flame of desire, nor are they set alight by the heat of lust."
A peacock with its tail down (rather than displayed) can be a sign of humility, of restraining one's pride as the peacock resists displaying its tail.
Apes represent the mockers
On/from the left or right hand
"The left hand represents temporal possessions; the right, eternal life. Those who manage temporal possessions sit on the left. Those who desire eternal life with all their heart fly on the right."
The turtledove symbolises fidelity, for it is believed that the female dove will not take a second mate if her first mate dies. Parallel are drawn to virtuous widows remaining chaste and single, thus a single turtle dove is a symbol of chastity (and possibly purity), and a pair of turtle doves has some symbolism of love. 
As the phoenix rises from the ashes, so is it a sign of the resurrection of Jesus Christ. This particular symbol was specifically associated with the resurrection of Jesus - in other times, it held a more general meaning of renewal.