What is the name of the finned metal device that radiates heat away from the processor?

This is called a "heat sink", essentially, a metal plate makes indirect (its separated by thermal paste) contact with the heat spreader of a cpu (central processing unit). The concept is that the heat sink is made of highly conductive materials thus conducting heat away from the CPU, quite often you'll find a fan attached to further disparate heat away from the heat-sink. These days you'll alternative cooling methods, the most common alternative is to use watercooling (another topic). GPU (graphic processing units) also have heatsinks.    

Just take a picture from your mobile.