Native plants are plants that occur naturally in your region. For example, Douglas-fir is a native plant in much of Western Washington. English holly, on the other hand, is not a native plant in Washington because it was originally brought here by humans (it is, however, a native plant in England).
Non-native plants are often called "exotic plants" or "introduced plants." Occasionally they can become a problem, spreading aggressively and damaging wildlife habitat.
The plants native to your region have grown alongside the native insects, fungi, plant diseases, wildlife, and other native plants for thousands of years. This long-time association has produced a complex web of interrelationships, by which the native plant may depend upon numerous other native organisms to survive and flourish, and a multitude of native organisms may, in turn, depend upon that native plant to survive.
In the process, native plants have evolved the ability to attract native animals that benefit them (such as pollinating and seed-dispersing insects and birds), and repel or survive native organisms that harm them (such as plant viruses and munching insects).
As a result, native plants often attract a wider variety of native animals than do exotic plants. In addition, the plants native to your area are adapted to growing in your region's soils and climate, and so generally require less maintenance (such as watering) than do non-natives.
Using native plants raises important issues about exactly what "native" means. For example, red-osier dogwood (Cornus sericea), is native to Western Washington. However, it is also native to a number of other places, including Alaska, southern California, Michigan, and Maine! Although they are all the same species, red-osier dogwoods growing naturally in other areas have adapted to a very different combination of climate, soil, diseases, and other plants and animals from what is found in Western Washington. As a result, you could say that dogwoods native to Michigan are about as "native" to Western Washington as are palm trees!
Ideally, you want to use plants similar to those that occur naturally nearby. Such plants will be adapted to the climate and soils specific to your area. In addition, using truly native plants will protect local native plants from crossing with similar plants from other regions (which can water down the local adaptations native plants have evolved over time).
Unfortunately, the red-osier dogwood sold here in nurseries frequently has been propagated from plants adapted to growing on the East Coast. Since most nurseries do not track the origin of their stock, it can be difficult to know what you are getting, and you may prefer to go to a nursery that knows their stock is from our region, or grow your own.