There's a few terms when it comes to plants (Native, Non-Native, Invasive) which are defined by USDA fairly well but for a quick and dirty: Native been in the same geologic region/evolved over time; Non-Native evolved in other parts of the world or cultivated and doesn't exist in nature naturally; Invasive, a non-native which has the ability to spread quickly and destroy the localized ecology.
Unfortunately, there's far more non-native and non-native invasives in our roadsides, our forests and definitely at our homes. They aren't going away - after all they are evolving to stay longer - so it's our responsibility to not only stop planting them but to remove them when/where we can.
Exotic invasives are detrimental for a variety of reasons including (but not limited to), destruction of habitat, suffocation/strangulation of large wood plants (both trees and shrubs), harbinger for ticks/rodents, erosion of soils, create habitats for destructive insects, and falsely offer insect larval planting but unavailable nutrients.
Larger Seeders
Tree of Heaven
Knotweed
Butterfly Bush
Barberry
Burning Bush
Bradford Pear
English Privet
Climbers
English Ivy
Porcelain Berry
Kudzu
Periwinkle
Japanese honeysuckle
Asiatic Bittersweet
Japanese Wisteria
Wintercreeper
Wisconsin Department of Natural Resources - Invasive Species Information
FoHVOS Strike Team
Pennsylvania Department of Conservation & Natural Resources - Invasive Plant Fact Sheets
Brandywine Conservancy - Invasive Species Spotlight
Butterfly Bushes are NOT Monarch Flowers