In mathematics, a singularity is a point where a function "goes to infinity".
A pole is an example of a singularity of a function defined on the complex plane.
For a function defined on the complex plane, an essential singularity exists at a point if and only if, for any real number R and complex number Z, there will be a point no farther than R from the point of essential singularity for which the function has the value Z. In other words, no matter how small a region around the point of essential singularity you take, you will still find all of the complex numbers inside it as values of the function.