Why do you need to refrigerate bread dough overnight? I'm making a no-knead bread (actually my very first time making bread!) and right after mixing the dry ingredients with water, it's supposed to sit for a couple hours to rise and then be placed in the refrigerator for some long period of time. Well wouldn't you know, after mixing the ingredients I accidentally let it sit all night - so it sat at room temperature for some 8-10 hours before I realized it. As soon as I found it in the morning, I put it in the fridge, where it has sat for now about 9 hours.

The dough's consistency seems fine. Is there any reason I shouldn't go ahead and bake it? (after bringing it back down to room temp, as the recipe calls for) Why does it need to be refrigerated after rising at room temperature for a couple hours?
The reason for doing delayed fermentation in the fridge is that the yeast development is slowed down, while still allowing the enzymes that naturally occur in the flour to do their work (converting starches to sugars, making a more flavorful dough).

The risk of doing it on the counter instead of the fridge is that the enzymes are working AND the yeast is working, which can over-leaven the bread, and they'll eat up the sugars you are trying to create with enzymes. Over-leavened bread isn't ideal, but it isn't the end of the world either. There is a lot to learn about bread baking if you want to really excel at it, but "just bake it, it'll be fine" is always present as you experiment - glad the loaf turned out well!

"No-Knead" bread is a great introduction to "rustic" doughs (very high water content), and to cold fermentation. Cold fermentation is about the best thing you can do for your bread as a home baker, whether or not you are using the no-knead recipe - I cold ferment nearly everything I bake.

If you want to learn more about the process and jump off the deep end, Peter Reinhart's book "The Bread Baker's Apprentice" is a great place to start. It is detailed without being pedantic, simple without being over-simplified, and oriented for the home baker without forgetting the reasons that professionals do things the way they do.