What are the whites of your eyes for?
The white part of your eye is called the sclera. It's made up of thick connective tissue, primarily Type I collagen, the same protein that makes up your tendons and ligaments. It holds the shape of the eye, and provides protection for the delicate inner structures of the eye.
But why is it so big and white?
To perform its functions, the sclera doesn't need to be nearly as big as it is. And it certainly doesn't need to be white. Take a look at your pets, or at pictures of animals. Compared to ours, their sclera compose only a tiny portion of the visible part of the eye. In some species, the sclera are disguised, with a color similar to that of the iris. In fact, no other species has sclera as big and distinctive as ours. Even monkeys have much smaller and darker sclera. And since the larger sclera don't make the eye any better at seeing, it seems there must be a non-visual reason for their distinctiveness.
It's thought that the sclera are so obvious in humans for a social reason. Having such large sclera allows other humans to see where we're looking with a high degree of accuracy. In other words, the whites of your eyes are there to allow you to broadcast your thoughts (well, some of them) wordlessly to other people, by making it obvious to them what you're looking at. Are you looking into their eyes, or just slightly to the side? Are you looking intently at an object, or just staring off into space?
Your eyes are able to make everyone around you a little bit telepathic.