Put simply, a weed is a plant out of place. Whether it's a beloved native species like aspen, that sends up little shoots everywhere, a despised invasive like broadleaf plantain, that spreads through the lawn, or even a garden herb gone wild like mint, that refuses to be contained, weeds are just plants in places we don't want them.
Generally speaking, "weedy" plants tend to be fast growing, creeping, highly reproductive, or otherwise quickly spreading. Those that are particularly troublesome tend to have long-lived seeds and/or deep roots, allowing them to reseed or re-sprout, even after they've been "eliminated". Weedy plants also tend to thrive in relatively disturbed soils, which happens to mean just about anywhere humans have been. Lawns, gardens, roadsides, fields, abandoned lots, mine tailings, etc., end up being hotbeds for weeds. Our yards, in particular, with a decades-long legacy of disturbance, including soil compaction; repeated introductions; chemical fertilizers, herbicides and pesticides; and a vastly different water regime; are fertile breeding grounds for weeds.
However, all native plants, weedy or otherwise, have lots of natural enemies, be they bugs, diseases, competitors, or deer, to keep them more or less in check. Non-native species, on the other hand, have left most of their natural enemies behind. Many introduced species, from roses to peaches, are nonetheless still quite well behaved. In the case of many garden edibles and ornamentals, their good behavior is largely because they've been bred by humans to be showy or tasty, not competitive. Other plants like Kentucky bluegrass, the species used in most lawns, are from much warmer and wetter climates, and barely survive here as it is.
"Invasives" meanwhile, are aggressively weedy, non-native species, that are well adapted to Colorado's climate, but have few natural enemies here. Oddly enough, many of the most pesky non-native garden weeds, like dandelions, aren't really very troublesome in the wild, because they don't do well in Colorado's dry, infertile soils. Others, like myrtle spurge or russian olive, which seem perfectly well behaved in a moist, fertile garden (and are often even planted deliberately as ornamentals!), spread like wildfire in native ecosystems. Still others, like the infamous field bindweed, are infuriatingly weedy invaders in both yards and native ecosystems alike.
So why should you care about invasive weeds anywhere other than your yard? They invade agricultural fields, causing increased herbicide use, increased herbicide runoff, poorer water quality, lower fish stocks, more health issues, lower crop yields, and higher food prices. They invade rangelands, causing decreased forage quality, lower meat yields, and higher food prices. They invade riverbanks, causing salinization of the water table, reduced water reservoirs, and changed erosion patterns. They invade wildlands, causing loss of biodiversity, loss of habitat and forage for game animals, desertification, topsoil loss, increased wildfire frequency and magnitude, and a loss of intrinsic aesthetic value.
