We blink to spread tears across the eyeball, and to remove irritants (like dust).
Although these two functions are logically distinct, they are also related - spreading the tears across the eyeball helps to remove the irritants, much as you get much more dirt off you car's windscreen if you wipe it after you've wet it.
So much for removing irritants - the reason for that function is obvious. But why is it important to spread tears across the eyeball?
The cells of the cornea (the clear bit in front of the pupil) are avascular; that is, they are not supplied by blood. How do they then survive? Well they depend on nutrients supplied by the tears. They also depend on the tears to keep them moist, because they are very vulnerable to drying out, since they are constantly exposed to the outside world. (Skin copes with this problem by coating itself in a thick waxy layer, but this is hardly an option if we still want to use our eyes for seeing!)
Furthermore, tears lubricate the eyeball, accounting for the fact that there isn't a horrible grating sound every time we try to blink. This lubrication also has the added benefit of reducing the tears' surface tension, ensuring that they are evenly spread across the entire eyeball, rather than forming little droplets with dry areas in between (as water usually does).
Lastly, tears also contain various substances that serve as disinfectants, helping to avoid eye infections.
That answers the question, but I can't resist quickly explaining the 'structure' of tears, because it's really clever. The inner substance of the tears is the aqueous layer, which contains the nutrients and 'disinfectants'. This layer is sandwiched between two other layers. The outer one is an oily lipid layer. It provides the lubrication function, but there's more to it than that. I don't know about you, but on the odd occasion when my eyes were dry, I tried going to the bathroom and splashing some water in them. It doesn't work. The water evaporates within seconds, and my eyes feel drier than before. The lipid layer is what is known as 'hydrophobic' (literally, scared of water), and it deals with its aversion to H2O by getting as far away from it as possible. It thus forms a film out the outside of the aqueous layer that greatly retards evaporation. It also prevents the tears from rolling down onto your cheek (unless you overwhelm the system by crying).
On the other side of the aqueous layer (on its inner aspect) is the mucus layer. It sticks to the cornea (and the conjunctiva, the thin layer covering the 'white' of the eye). Unlike the lipid layer, the mucus layer is 'hydrophilic' - it enjoys bonding with water. As a result, it ensures that the aqueous layer (and thus all aspects of the tears) is evenly distributed across the eyeball.