What is this spoon-like kitchen utensil with a rubbery head and wooden handle called?
That is usually called a rubber spatula, even though they're rarely made of rubber. The ones I find most useful are made of silicone, and are heatproof. This one by OXO is one of my favorite gadgets.

They're for stirring and for clean scraping bowls and jars and such. The heatproof ones are great for use with non-stick cookware while sauteing.