Non-native species are species that have been introduced in a certain area. They do not historically belong to that area. Therefore, introduction basically means the transfer by humans of an organism outside its native range.
In contrast, "native" species are species that have historically occurred as part of an ecosystem in a specific location.
For many centuries species have been introduced into new areas, mostly for economic benefit, but sometimes by accident. Very often an introduced species goes unnoticed until the population has grown so large that it might be very expensive, difficult or impossible to eradicate the species from the area.
When species live in their natural habitat they have to compete for the same resources with other organisms that live in the same habitat. For this reason the species number will remain within normal proportion. But when a species is released into a new area, the forces that control the species number are different and the species may start to increase uncontrolled.
When this happens the species is competing with native populations or native species and this eventually may be to the detriment of the native species. Then the introduced species is called an "invasive" species.