What is the difference between a singularity and a pole?

One says that $z_0$ is an isolated singularity of $f$ if $f$ is defined in a punctured neighborhood $D\setminus\{z_0\}$ of $z_0$.

One says $z_0$ is a removable singularity of $f$ if there exists a holomorphic function $F(z)$ defined on $D$ which extends $f$.

Suppose $f$ is nonvanishing in a punctured neighborhood $D\setminus\{z_0\}$ of $z_0$. Define $F(z)$ on $D$ by $F(z) = 1/f(z)$ if $z \neq z_0$ and $F(z_0)=0$. Then $z_0$ is a pole of $f$ if $F$ is holomorphic at $z_0$.

A singularity $z_0$ is an essential singularity of $f$ if $z_0$ is neither a pole nor a removable singularity.

A pole is a (usually defined as a) special case of a singularity. See http://en.wikipedia.org/wiki/Pole_%28complex_analysis%29