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 a value within R of Z. In other words, no matter how small a region around the point of essential singularity you take, you will still find all, or nearly all, of the complex numbers inside it as values of the function. (A function can miss two complex numbers; e.g. tan(1/z) never quite reaches +/-i.)