Touring bike or a mountain bike. If you want something sportier, you can try a cyclocross bike (a heavily modified road bike designed for offroad use as well as on).

Road bikes are designed for paved roads, as opposed to gravel. They can do it, but not near as greacefully as a bike with larger tires can.

Also get good tires.....tell the shopowner that your first priority is flat resistance. They should offer up either a continental gatorskin or a specialized armadillo tire (Serfas with is fine if they have none of the other two, so long as it's one of the models with flat protection...I'm a serfas fan myself). Get those, along with tire liners, and a "thorn resistant" tube (thicker on the tire side than normal tubes, so even if a thorn gets through, it has to go through more rubber in the tube as well).