Bikes that work on both paved and non-paved surfaces can include mountain bikes, cyclocross bikes, tour bikes and hybrids. Tour bikes are essentially a variant of road bikes that can accomodate slightly fatter tires, and racks and fenders. They can handle moderate off road conditions. Cyclocross bikes are road bikes adapted to racing on mixed surfaces, and they can also accomodate the fatter tires, and often racks and fenders as well.
Hybrids range from upright cushy comfort bikes that are quite slow, to lean and mean flat bar road bikes. You'd have to look at specific models to see if they suit you.
If you want a mountain bike that also is fast on the road, get a rigid model without suspension (you may have to go used, from the 1990s) and put thinner, slicker tires than usual on it.