jennifer....
If you google: "Suckers, and Water Sprouts" you'll find some photos of exactly what these "branches" look like...and, an even more explicit, G-rated version can be found by going to: Newton Tree Conservancy....hit: "News and Events"...scroll down and hit: Citizen Pruners Training Events Spring 2009....you'll see just what a sucker is and how to remove this unwanted "thing". You will not have to avert your eyes, nor will you get "the vapors". Put the Feinting-Couch in storage, and whip out your pruning-shears. : ) : ) Very visual, picture heavy...
Hope this helps.
Frank