Noxious weeds are plants designated by weed control organizations as legally requiring treatment when they are present. It is the landowner's responsibility to eliminate them from their property and fines can be assessed for failing to do so.
Typically noxious weeds are non-native invasive plants that were somehow introduced and have no natural enemies to keep them in check, as they did in their native environments. They prevail in the environment and push out the native plants -- ruining grazing for wildlife and impacting agricultural operations. Studies are underway to determine the impact of these noxious weeds on ground water depletion -- it is substantial.
"Invasive plant species are estimated to cause more than $20 billion annually in economic damage and affect millions of acres of private and public lands." (ASI Grazing Workshop ~ Spring 2008)
For more information on your local noxious weeds: