Invasive plants are aggressively growing plants that rapidly spread and displace other plant material. Some invasive plants are worse than others. The biggest issue is that some homeowners either don't know that the plant is invasive or they just don't care. Just because a plant looks really pretty does not always mean that it belongs in your garden.
It has been said that any plant that is not native to that region is invasive. This however has caused some great debate. Any plant that is not native was either accidentally introduced via agriculture or was purposely introduced. Some non native plants are not highly invasive, only colonize small areas and are not aggressive. However there are some non native plants that are really invasive and disrupt many natural habitats. These highly invasive plants pose a great threat to the natural ecosystems.
As with any plant that one chooses to use in the landscape it is best to first consider the ones that are native to your region. For almost every non native plant there is an equally matching native species. Regardless of what you plant in your garden it is always best to do your homework. In fact the best design technique is to simply hire a professional landscape designer.
If you live in California and want further information on invasive plants then visit the California Invasive Plant Council website.
Note: the picture above is an invasive Yellow Star Thistle.