Plants can be problematic, too. Species such as multiflora rose, English ivy, Chinese Wisteria, Crepe Myrtle, Japanese honeysuckle and Bradford Pear have all escaped captive culture into the wild. In the wild they may or may not out compete native plants.
English Ivy, Wisteria, and honeysuckle will form dense mats of one species. These mats can strangle large trees and smaller native plants. They reduce the biodiversity of the areas they invade. Most of the time they don't benefit local wildlife and by smothering native plants they actually reduce the local food sources (although English Ivy will increase local rat populations since it hides them from predators). These monster plants are termed Invasive Exotics. Invasive is sometimes used by itself with the term Exotic understood. Exotic simply means they are not native.
Plants that are native but spread and take over (especially in the garden) are termed aggressive. Many gardeners will use the stronger term invasive to describe these plants but that is kind of like referring to every cold as the flu. The flu is a specific infection, not all head colds are the flu.
Even the term native is not always clear. What time period are you referring to? Pre-Columbus? I'm sorry but no one got off of the first ships here and did an intensive inventory of the native plants and animals so we really don't know what was here before Europeans. And if you want to go back before Native Americans well, that was a very different America. In fact I don't mean to shock you but we still don't have a complete survey of life forms in America. Check out www.dlia.org/ for information on the Smokey Mountain Biodiversity Inventory Initiative that is attempting to correct this for a small part of the Great Smokey Mountains. They have found plenty of species previously unknown to science just in the past few years.
OK, so let's set a random date and say all species that occurred before 1800 will be called native. What about the species that showed up here without help from humans since then? Two species of Ibis, the Glossy and the White-faced Ibis, both arrived on the shores of the US without help (probably blown across the Atlantic with stormy trade winds) and have since spread completely across the country. So are they native? Or are they exotic?
Let's recap. Invasive is a term that refers to a plant or animal that escapes (or is released) into the wild, except by some people who use it to refer to species that take over the garden but wouldn't survive in the near-by semi-wild lands. Aggressive is a descriptive word that means a plant that out-competes near-by plants, whether that occurs in the garden or the wild. OK, that's clear. You can have an aggressive invasive species (like Chinese Wisteria) or a non-aggressive invasive species or even an aggressive native species. Whew.
Then there is the native issue. That word sounds good and is generally used to refer to plants and animals that occur in the wild or semi-wild, without human assistance. But some things, like Pigeons and Starlings, do quite well without human assistance and are definitely not native species. Again it's not too clear cut. And that doesn't even begin to address the issue of species that have shown up in the US without human aid or have spread into new parts of America in the recent past (think Coyotes).
So we have descriptive terms that are not clear cut to describe problematic or potentially problematic species. Why does this matter? Because I believe that as Gardeners we have a responsibility to the local wildlife and ecosystem to do no harm. There are not many truly wild spaces left and all kinds of species, butterflies, birds and wildflowers are in decline due to habitat degradation and human expansion.
There is plenty of evidence that wilderness is valuable to the human psyche so even if you don't value it for it's own sake you should value it for yours. Not to mention the damage to our air and our water supply if we neglect our own ecosystem.
One good thing for gardeners to do is to avoid planting plants that have a tendency to escape and aggressively out-compete native plants. These are the plants that are usually categorized as invasive exotic species. There are various, regional lists out there of the worst bad guys. In North Carolina the North Carolina Botanical Garden has some great information about good and bad plant choices for the Southeastern US (http://ncbg.unc.edu/plants-and-gardening/), and they are a great source for native plants. In New England the New England Wildflower Society talks about them at http://www.newfs.org/protect/invasive-plants. Lastly the USDA also keeps track of species that they consider native and invasive. You can enter individual plants into their databases to find out if that species is problematic in your area. You can find this database as http://plants.usda.gov/.
So why, you ask, did I include a photo of the World's Cutest Cat, Katydid? Guess what beloved pet species can be aggressive, even invasive, when accidentally or on-purpose released into the wild? Katydid is quite happy inside, making sure her blankie doesn't get stolen.