It does look like fun, doesn't it?
Cyclocross bikes look a lot like road bikes (drop bars and all that), but have wider tires and a few other tweaks to make them mud-friendlier. Some have disc brakes. IIRC, Specialized makes the Crux and Tricross. Giant has the TCX, and there are others (Surly Crosscheck, Redline, etc). You'll have to do some more research if you truly need a WSD bike, because there aren't that many WSD cross bikes out there.
And yes, you can absolutely ride them on the road--just swap the tires out. Also good for winter because they can take studded tires--good if you live in the frozen north. If you're only going to have one bike, a CX bike is probably a good choice!