Invasive plants are alien species that show a tendency to spread out of control. Although not synonymous with "exotic plants," the "invasive" label is now typically reserved for plants that have been introduced from other regions and spread like wildfire in their new habitats. Indigenous plants that spread rapidly and overpower the competition tend to be referred to now as simply "aggressive," "thuggish" or "ill-behaved," in common parlance.
It is thought that the tendency of invasive plants to spread so much may be due in part to the fact that the insects and diseases that plague them in their native lands are often absent (or exist in lower numbers) in their new homes, where the invasive plants thus enjoy a "free rein," relatively speaking.
Helping invasive plants spread in some cases are extensive underground networks of root-like plant parts called, "rhizomes." The rhizomes are so widespread that attempting eradication by digging them up is usually fruitless.
Invasive plants compete so successfully against other plants that they can crowd out their competitors, thus producing a monoculture that discourages the growth of other plant species. These exotics often specifically crowd out indigenous plants in this manner -- a fact that makes the "invasives" issue a hot topic in some circles (especially in the native plants movement). Classic cases of invasive plants forming such a monoculture can be seen in entrenched stands of Japanese knotweed and purple loosestrife (picture), both of which have notoriously vigorous rhizomes.
Landscapers need to act aggressively to eradicate invasive plants that invade the lawn or garden. Many would classify this activity as "weed control," but note that the terms "invasive plants" and "weeds" are not synonymous. Some noxious weeds happen to be invasive, but not all are. Nor are all invasives weedy-looking. Some are quite beautiful; I show examples in my photos of invasive plants.
My full article on invasive plants will also help you identify some of the worst offenders.