Not so very different from rose - its Rosa, followed by the varietal name, so one example would be Rosa Paprika.

They are mostly complex hybrids between a number of European and Asian species, so a full botanical name is somewhat problematical. The closest that I have found to a proper botanical name is "Rosa x hybrida", followed by "c.v.", for cultivar, followed by the variety name in single quotes--i.e., "Rosa x hybrida c.v. 'Peace'". A few species roses are still occasionally available, such as the Austrian Copper Rose (Rosa foetida fa. bicolor), and the Scotch Burnet (R. spinosissima). Here in the desert Southwest, Lady Banks' Rose (R. banksiae) is commonly sold as a wall- or arbor-covering vine--though it's rarely included with the hybrid roses.