There are some good long answers already, so I'm going to try to give as short an answer as possible.

A quantization of $X$ is some $X_\hbar$ depending on a parameter $\hbar$ (occasionally $q=e^\hbar$ instead) such that $X=X_0$ and $X_\hbar$ is generically "less commutative" than $X$.  This is by analogy with quantum physics where $X_0$ is classical physics and $\hbar$ measures the failure of position and momentum to commute.