You have a Rose Bush. However, the mossy looking appendage on your Rose is called a Mossy Gall. It is the plants reaction to an insect, disease or mite. The damage causes the plant to change the course of normal growth. It modifies growing tissue into a special swelling that surrounds the gall-maker. In the case of mossy rose galls it is a small cynipid wasp. This Rose has the scars from previous galls. There is no treatment once a gall has formed.
Your rose has dead wood that needs to be removed. Here’s some information on general rose care, including pruning tips – http://www.ext.colostate.edu/ptlk/1725.html
Here’s more information on Mossy Rose Gall – http://www.ipm.iastate.edu/ipm/hortnews/2008/7-2/mossyrosegall.html