Do Dogs Understand Us?

Better than I would guess:

But dogs do have many ways of telling us things. When a puppy wags its tail or barks or runs around in circles as we arrive home from work, we get the gist. I'm hungry. Let's go for a walk! Get off of my property. And mounting evidence shows that dogs understand human language better than previously assumed (except by dog lovers). They're about as smart as a 2- or 3-year-old child, the age at which most kids begin to initiate conversations and speak in simple sentences.
To acquire language, kids use a strategy called "fast mapping" -- forming quick, rough hypotheses about the meaning of new words after just one or two exposures. So do dogs. Recently, researchers found that a border collie named Rico was able to infer the names of more than 200 items using this method.
Four weeks after the initial exposure, Rico was still able to retrieve the items by name. Another border collie in South Carolina has memorized over 1,000 nouns. The dog, Chaser, reportedly loves her vocabulary drills.
A similar study out of the Max Planck Institute showed that puppies use human communicative cues to solve problems by as early as six weeks.

As smart as a 2 or 3-year-old, and at least as destructive.