Centering involves subtracting the overall sample mean score from the original score; standardizing does the same followed by dividing by the overall sample standard deviation.
Centering is the act of positioning something at the midpoint of a space. For two-dimensional spaces (lines), this usually involves taking the length of the line, dividing it by two, and placing the object at the point denoted by this value. For higher dimensions, the same technique is also applied for each additional dimension, as each each dimension can be considered discrete.
In many statistical procedures, it is helpful to center the mean of a variable's distribution. Examples include general linear models with interaction or quadratic terms.