Once trees are removed to ground level, the resulting stump is often required for removal too.  Smaller stumps can often be removed by using hand digging tools.   
 
Larger stumps however are usually best removed using a machine called a “Stump Grinder”.  Where access is viable, these machines seem to effortlessly remove the stump using a large cutting disc fitted with special cutters “teeth”.  The stump is removed approximately 300mm below ground level where possible.  Any remaining root and any stump will rot and break down over time.  Once the stump is removed “ground out”, the hole will be reinstated and backfilled, leaving a neat & tidy finish. 
 
But do you need to remove the stump?  If it is not posing a hazard in its location, then why not leave it?  Resulting stumps can be treated with a herbicide product to prevent regrowth and reduce cost.