We all know that the pupil of the eye is black. But that is a bit surprising; after all the pupil is simply an opening into the eye, so when we look someone in the eye, why do we not see the inside of it?
A first guess might be that the inside of the eye is in fact black. That is not the case, however: the retina is red-orange from the many blood vessels in it. Another guess might be that the pupil appears dark because the interior of the eye is dark. But the very purpose of the eye is to collect light, and its inside is almost as well-lit as the rest of the room it is in.
The true explanation is positively Hofstaderesque. The lens of an eye focuses all light from any given point in the room, for example an observer's eye, to a single point on the retina. So when you make eye contact with someone, you see only a single spot on his/her retina, and that spot happens to be completely unilluminated. Why is it not illuminated? Well, consider where a lightsource would have to be to shine on it: where your pupil is. And as we all know, pupils are black...
There is an instrument, the ophthalmoscope, to get around this problem so that the retina can be studied. It uses a half-silvered mirror or a prism to reflect light into the eye in a way that lets you look along the light-ray.
Update: Fruan has a more prosaic explanation; he says that he has dissected several eyes which were in fact black on the inside.