A noxious weed isn't just any plain ol' weed, nor is it necessarily a poisonous or inherently harmful one (although many are). In essense, a noxious weed species is non-indigeneous (which immediately renders the term relative to geographic location), has no natural predators, reproduces voraciously, and is thus horrendously destructive to the surrounding environment. Noxious weeds disrupt the ecosystems in which they are introduced, simply by competing heavily with local plant species. Since the local flora have natural predators and have established a sort of reproductive harmony with their surroundings, noxious weeds present a formidable threat to the native plantlife. Not only are local plant species affected, but wild animals and livestock suffer as a direct result (certain weeds are downright toxic to grazing animals, which are often oblivious to the foreign scourge's nasty nature). As if that wasn't bad enough, they can also increase erosion. Whereas many undesirable plants are merely considered "nuisance weeds," noxious weeds are blamed in causing billions of dollars in lost agricultural productivity every year.
Noxious weed species often originate from countries thousands of miles away. Here in Colorado, Russian knapweed is proving to be quite a scourge, along with its cousins spotted knapweed and diffuse knapweed. Not only does it reproduce in a destructively virulent manner, but it causes brain damage in horses to boot. So far, traditional measures such as spraying with herbicides and pulling by hand have proved ineffective, so five different insect species are being reared for the express purpose of eradicating them. Other methods of control include fire, tilling, and seeding the area with indigenous species to stress the invaders.
A list of noxious weeds (from Colorado's perspective) can be found at http://www.ag.state.co.us/DPI/rules/noxious.html. Of course, the definition of "foreign" varies everywhere, so one man's noxious weed could be another's ornamental. Some of the more notable entries include:
African rue aka Syrian rue, a plant rich in harmala alkaloids, including the MAO inhibitor harmaline. Syrian rue is said to both prolong (by as much as 2x) and intensify the effects of tryptamines such as psilocin, the active ingredient in psychedelic mushrooms of the Psilocybin and Stropharia genera. Actually, MAO inhibitors will intensify the effects of just about anything, usually to deleterious extremes (e.g. death by cerebral hemmorhage). Mixing cheese, certain kinds of fruit, or even chocolate with an MAOI can be dangerous.
Chicory, a popular "herbal alternative" with diuretic and laxative effects.
Common St. John's Wort, yet another "herbal alternative" touted for its prozac-like effects (although some studies are casting doubt on its efficacy and safety).
Common mullein, also affectionately known as "Indian toilet paper" (no offense, I'm aware of the proper term "Native American") due to the inherent softness of its leaves. My grandmother purported to have attempted to utilize it for this purpose in her childhood, and would not recommend it. Those hairs might seem soft when they're all packed together on the leaves of the plant, but it's a whole different story when they're embedded in your sphincter. I see this one all over the place in Colorado, as it's quite easy to recognize.
Poison hemlock, which is toxic to just about everything. Symptoms of ingestion include loss of coordination, nervousness, coma, and death.
Mediterranean sage, an edible herb of the Salvia genus.
Wild mustard, yet another edible herb, the leaves are perfectly acceptable greens.