I've hear that Ducati is a motorcycle manufacter based in Bologna, Italy.
I've never heard of them. Have they been around for a long time?
it seems the company was founded in 1926 by Antonio Cavallieri Ducati and his three sons
Are these motorcycles sold in the united States?
Yes, they are.  The comany is now owned by German automotive manufacturer Audi and its Italian subsidiary Lamborghini.
Are they for on the road or are they used for racing, or both?
I am not sure, but I think probably for both.
I wonder how much one of these things costs. More than a Harley?
Maybe, since they probably are imported and the Harley-Davidson is an American manufacturer.