Picard's Great Theorem
Every non-constant entire function attains every complex value with at most one exception. Furthermore, every analytic function assumes every complex value, with possibly one exception, infinitely often in any neighborhood of an essential singularity.
Can someone give me some concrete examples explaining this result?
Fort the first part, the function $f(z)=e^z$ is a typical example. It is a non-constant entire function attaining every value with one exception - it is never zero. For the second part, typically $e^{1/z}$ is considered. One shows that arbitrarily close to the essential singularity $z=0$, all non-zero values are attained. You can "see" this in the plot here: http://en.wikipedia.org/wiki/Picard_theorem.