What is a surjective homomorphism of algebraic groups?

Question

Question is as in the title. I know this is a very basic question, but I couldn't find a definition anywhere. I thought it means that the morphism of varieties is surjective, and I think that should imply the morphism is also surjective on functor of points (correct me if I'm wrong!), but I saw somewhere that the morphism $\mathbb{G}_m \xrightarrow[]{x \mapsto x^n}\mathbb{G}_m$ is surjective but $\mathbb{G}_m(\mathbb{R}) \xrightarrow[]{n}\mathbb{G}_m(\mathbb{R})$ is not.