I notice when I bake cookies (usually chocolate chip or peanut butter) the recipe often says "place on baking rack to cool". Because I don't have one, I just place them on a plate after removing them from the pan. It doesn't seem to do any harm. What is the point of buying this special piece of equipment?
A cooling rack serves two primary purposes.
First, it allows the cookies (or other baked good) to cool faster by letting air circulate completely around the cookie.
Second, it prevents the steam escaping from the cookies from soaking the bottoms, and other cookies placed on top.
In addition to hobodave's cookie-specific answer, as a general point (since this was asked as a general question) it's also quite important to have a rack if you are baking anything with a significant quantity of fat (i.e. meat). The rack allows the fat to actually drain; otherwise the food would end up being drowned in its own fat and become soggy. In extreme cases the fat may actually burn and seriously taint the flavour of the food you're baking.
When you place something on a cooling rack you expose more of the surface area to the air allowing your cookies, bread, etc to cool faster than just leaving them out on the pan.