In heraldry and other symbolic use, the beasts of nature have long been powerful symbols. The presence of a lion rampant or an eagle with its wings spread on a shield shows the bearer's courage, bravery, and more. Many animals have been used as emblems in this way, each with their own symbolic meaning and connotations.
The lion is one of the most popular images for heraldic use, and has showed up in statuary, art, and more with similar meanings. As the lion has been regarded as the king of beasts for centuries, thanks to an early bestiary known as The Physiologus, it is a symbol of royalty or nobility, often shown crowned or otherwise garbed with regal attire. For obvious reasons, it also has strong connections with strength and bravery, being a symbol suitable for a great warrior. Lionesses are also often connected to fierce protection of culture, family, or any similar group. The lion rampant (on its hind legs) brings to mind these connections with strength and fierceness.
The eagle, much like the lion, is a noble symbol, being thought of as the king of the air. It has particular connections to courage and foresight, and is thought of as a messenger of the Gods. Christian art often connects it to St. John the Evangelist, matching its image as a messenger of the divine. An eagle displayed with its wings spread is a mark of protection and shielding, and one who bears the symbol of an eagle is thought of as courageous and judicious. Those countries that used a double-headed eagle used it for one of several reasons - it could represent dominion over both the secular and the religious or both the east and the west, or be used as a symbol of the division and unity of church and state. Interestingly, for some time the major noble families of Europe used either a lion or an eagle, with the choice often influenced by blood connections, and so conflicts were often divided between the king of the beasts and the king of the air.
Many animals, while used less commonly, still have important connotations in heraldry. The bear, much like the lioness, is associated with defense of family, along with cunning and strength. The wolf, while less commonly used due to its nature as a predator of livestock, was considered courageous and often connected to the rewards of persistence and hard work. The pelican is interesting as it is nearly always depicted as piercing its own breast to draw blood to feed its chicks, described in heraldry as the bird being in her piety. This image of self-sacrifice has religious and spiritual connotations, and the bird is rarely depicted in any other position. If a boar is depicted in full, it symbolizes the positive qualities of the animal - courage and strength in battle - but if just the head is depicted it means that the bearer is a great hunter or hospitable to guests.
Many other animals are used in symbolism and heraldry, of course, but the lion and the eagle are easily the most predominant - as the kings of their domains, they of course take precedent. Still, every animal has its own unique symbolism and its own meanings.