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Data transformation is the mathematical re-expression, often nonlinear, of data values. Data are often transformed either to meet the assumptions of a statistical model or to make the results of an analysis more interpretable. Specific types of transformations are covered by the normalization and standardization and boxcox tags on Cross Validated.

1. Linear transformation. A linear transformation preserves linear relationships between variables. Therefore, the correlation between x and y would be unchanged after a linear transformation. Examples of a linear transformation to variable x would be multiplying x by a constant, dividing x by a constant, or adding a constant to x. The most common linear transformations are centering and standardizing.

2. Nonlinear transformation. A nonlinear transformation changes (strengthens or weakens) linear relationships between variables and, thus, changes the correlation between variables. Examples of a nonlinear transformation of variable x would be taking the square root of x or the reciprocal of x.

   In regression, nonlinear transformations of the response variable are sometimes used to achieve homoscedasticity and / or make the distribution of the residuals more normal. For example, here are a few methods of transforming variables sometimes used in regression settings:

normalization refers to scaling all numeric variables in the range [0,1], such as using the formula: $$x_{new}=\frac{x-x_{min}}{x_{max}-x_{min}}$$

standardization refers to a transform to the data set to have zero mean and unit variance, for example using the equation: $$x_{new}=\frac{x-\overline{x}}{s}$$

boxcox refers to the use of power transformations developed by statisticians George E. P. Box and David Cox in 1964 to identify an appropriate exponent, $\lambda$ (i.e. $Y^{\lambda}$), on the range of $[-5,5]$. For $\lambda=0$, the data is transformed by $\log_{10}$. Therefore, for $\lambda=2$ the data $Y$ is transformed by the equation $Y^2$

Reference: stattrek.com