The White Rabbit is a bipedal white rabbit who and talk and act very mush human. He is the first character Alice sees and she follows him down the "rabbit hole" to reach the world of Wonderland.
The White Rabbit was created by Lewis Carroll and first appeared in the novel Alice's Adventure in Wonderland in 1865. The White Rabbit is a public domain character like of of Carroll's characters from Alice's Adventure in Wonderland and the sequel Through the Looking-Glass. These two books are considered canon today. Everything that came later was some sort of sequel, continuation or behind the scenes kind of adaptation. Sometimes the stories are expanded as a what really happened kind of book.
Lewis Carroll Canon
The Rabbit in the original two children novel is an opposite of Alice herself. He bases everything on status. Anyone who he sees as below him he is rude too. While he grovels to his superiors.
The White Rabbit was just an ordinary rabbit when scientist Henry Allen purchased him at a pet store for use as test subject (circa 1860). He was the first of two test subjects (the second being the Cheshire Cat) to be used in Allen's experiments with the portal to wonderland. The rabbit is forced through the portal by Allen and transported to Wonderland. There the Jabberwocky transforms him, by twisting his innocent nature into its opposite: pure mad corruption.
After her suicide attempt, Alice Liddle receives him as a gift to comfort her-- seeming an ordinary rabbit again at that time. He lures Calie Liddle into Wonderland, where she observes his true grotesque form. With Calie trapped in Wonderland the rabbit returns to Alice, whom he slowly drives insane and leads to her eventual suicide.
League of Extraordinary Gentlemen
In the League of Extraordinary Gentlemen Universe Wonderland is a much more dangerous place. Alice doesn't even survive long after her experiences. The White Rabbit can be seen stuffed and behind glass in the League Museum.
In an obvious spoof the character of the Rabbit is played by a young girl. She dresses like Lewis Carrol's White Rabbit but refers to herself a bunny. The only real plot to the spoof continually leads to sexual acts.
The White Rabbit appears as a Fable on the Farm.
Lewis Carroll Canon
The Rabbit's powers so-to-speak are he is a man/rabbit he seems to be a sentient rabbit walking on two legs and being able to talk.
The rabbit is able hide his decayed and rotten appearance in favor of the appearance of an ordinary white rabbit, as he was before his trip to Wonderland. He may also communicate telepathically. Like all denizens of Wonderland he is ageless and gains increased healing abilities while in Wonderland.