The impacts of introducing a non-native or invasive species to an ecosystem will vary depending on a number of factors. Sometimes, the introduced species may not survive, if there is no ecological niche for the species to fill or the species cannot adapt to fill a different ecological niche.
However, if the species is able to thrive in a variety of environments and consume many food sources, that species will likely do well. No two species can share the same ecological niche, so either the native or non-native species will be better adapted and survive. If the invasive species is better adapted, it will out compete the native species. If the species reproduces quickly, it is also more likely to thrive in a new ecosystem.
Typically, invasive species harm an ecosystem, while non-native species either have a positive impact or have no impact. Therefore, if the introduced species adapts to fill a niche that is not already taken, it won't change the balance of the ecosystem; and maybe even help. If it competes with another species and succeeds in filling it's ecological niche, then the introduced, now invasive, species can harm the ecosystem.