One of the most common sources of inspiration for superheroic characters, aside from pre-existing superheroes in comics, is the animal kingdom.
And yet, creating an animal-themed superhero isn't necessarily as simple or simplistic as choosing a cool animal and naming your character after it. There are many approaches to animal-themed characters, but we can group them into three somewhat loose categories: animal-inspired, animal-infused, animal-inherent.
One good example of this is Batman, who didn't gain powers after being bitten by a radioactive bat, but instead took on the look of one in order to scare criminals (a superstitious, cowardly lot). Aside from the general bat-like look, the cave and the occasional bat-named gadget, the character is really more of a dark knight detective / caped crusader.
Another somewhat surprising example of this would be Wolverine. This short and ornery mutant with adamantium claws and a healing factor doesn't seem to have taken much from his namesake (wolverines are essentially badgers with worse attitudes) aside from being irascible and having claws. To top it off, he is sometimes shown in pictures alongside wolves -- which of course are not wolverines.
Spider-man is the classic example of this, thanks to that irradiated spider that bit him and passed on abilities that could arguably be considered as spidery abilities: wall-crawling (cool), proportional strength (ok, sure), spider-sense (uh, what?). Furthermore, he completes the ensemble with invented web-shooters and web-formula to show his dedication to the spider theme.
Hawkman can sometimes be considered to have this, depending on the incarnation of the character in DC Comics. At one time, not only did he fly and have keen eyesight -- he also had the ability to talk to birds! Of course, his huge arsenal of weapons borrows more from the metaphorical connotations of hawks (war-oriented), and he sometimes comes across like Conan with wings, but there's no denying the strong infusion of "hawk-ness" in the character.
Spider-man, at certain points in his career, became like this -- when he sprouted extra limbs for example.
The Teenaged Mutant Ninja Turtles are another example of this, having been regular turtles before a radioactive isotope that blinded a lawyer further up the street found its way to them and mutated them. While it may be argued that ninjitsu and a fondness for pizza are not very turtle-like, there's considerable weight on the turtleness to keep them in this category.
Kemlo "Hyperdog" Caesar from Alan Moore's Top 10 is another example of an animal-inherent character. In fact, he's essentially an uplifted dog. Out of his suit -- which lets him walk around, manipulate things with fingers, and generally look like a human with a head of a dog -- Kemlo actually seems to be just that: a dog (who's really smart and can talk).
Lycanthropes fall into this category as well, naturally. In fact, it seems that the battle between human nature and beast nature is a common trope of this type of character.
Beyond these categories
However, there are metahumans that are able to draw from multiple animal abilities and can be considered to draw from all or none of these categories: the meta-animal superhumans.
Beast Boy / Changeling was able to shapeshift into different creatures from the animal kingdom. Early versions of him had him with the natural form of the animal with green-faced head grafted on top of it. Later versions of him had him look like a normal -- albeit emerald-green colored -- animal.
Animal Man was able to draw upon the abilities of any nearby animal. He could fly if near birds, run a proportional speeds if near ants, gain the senses of various animals when investigating. In many ways, he was like Vixen, who used her Tantu Totem to also draw upon aspects of various beasts.
Last, but not least, there's B'wana Beast. Who had the ability to physically combine different animals together, and retaining the best abilities of each. Like horses and spiders, or sharks and pelicans. Wow, nothing creepy there at all.