Albino rabbits have red eyes because a lack of melanin causes the blood vessels in their eyes to be visible. Their eyes actually have no color and are clear.
Despite the common misconception, red/ruby eyed white, or REW, rabbits are not usually a breed unto themselves but rather a form of mutation known as albinism, in which the body lacks the proper genes to produce the protein melanin. Melanin is what gives the skin, hair and eyes its darker color in order to protect against harmful radiations from the sun. Because an albino rabbit lacks this protein, the eyes become clear and the blood vessels in the eyes are visible, giving the eyes a red or ruby look. There are also REW rabbits who are albinoid, possessing only some characteristics of albinism, usually manifesting in white fur with very light patches of dark spots and light-red or pink eyes. REW rabbits may come in all sizes and temperament because albinism is a mutation that happens across all rabbit species.
A notable exception is the New Zealand White rabbit, which has been specifically bred to induce albinism. Unfortunately, as a byproduct of albinism, REW rabbits are more prone to vision problems and some may be sensitive to sunlight.