Consider $f \colon {\mathbb C}^2 \to {\mathbb C}^2$ given by $(x,y) \mapsto (xy,y)$. The image consists of ${\mathbb C}^2$ minus the subset $y = 0, x \neq 0$. Since this subset is not closed the image is not a variety.
The notion of a constructible subset was invented to deal with questions like this. A constructible subset is one which can be constructed from subvarieties using "boolean operations". Equivalently it is a subset defined by polynomial equations and inequations. It is true that the image of a constructible subset is again constructible (Chevalley's theorem).