We blink because if we didn't, we'd all walk around staring at each other like a deer caught in headlights. Animals that don't blink are highly disconcerting to watch as well. For example, have you ever gotten in to a staring contest with a pigeon? Don't. You can't win. They don't have eyelids.
The eye is a fascinating structure. It is essentially a hollow ball with a lens in the front and a light receptor in the back. Light enters through the front and is focused on the retina on the inner rear of the sphere. Special nerve cells attach to the retina which transmits the light to your brain. These signals are then interpreted by your brain to form an image such as "That Creepy Pigeon with No Eyelids". Yikes!
The outer front of the eyeball is known as the cornea. The cornea is living tissue that serves as a barrier to protect the delicate inner structures of the eye from outside dirt, dust and other nastiness such as pigeon droppings.
The entire eye sits in the eye socket and is attached in the ear to the optic nerve. On each side of the eye are a series of muscles which are used to move your eyes around, allowing you to gaze in one direction, while keeping your head still. Imagine how inconvenient it would be to have to be looking straight at an object to see it? Leering at a pretty woman in a bar would be a lot more difficult. But I digress.
Because your eyes are delicate and somewhat vulnerable, they need to be protected and taken care of. This is where eyelids and blinking come in. Your eyelids are protection for your eyes. Think of them as "storm shutters" for your eyes. They block all manner of things from getting in your eye, as well as blocking light when you choose.
Blinking is your eye's way of turning on the windshield wipers, as you would in your car. The eye must stay moist in order to comfortably swivel around in the eye socket. In addition, there has to be some way of cleaning out particles and other junk that may get in your eye to cause damage.
Tears provide the lubrication, and blinking provides the mechanical wiping motion to clean out your eyes. The typical person blinks automatically every 4 to 6 seconds. If you try to "hold your blinking", you can do it for a while, but you'll soon notice that the eye becomes quite uncomfortable after a short time. This is because it's getting too dry.
So there you are - a long answer to a short question. Blinking is necessary to keep your eyes wet and to help keep them clean. Which begs the question, how in the world do pigeons keep their contact lens moist?