leighton provides an excellent description of an independent variable in the context of an
analysis of variance. In other contexts, however, an independent variable takes on a different character.
In many fields of scientific inquiry, the principal method of data analysis is not the analysis of variance, but is instead regression and its multivariate equivalents (such as canonical correspondance analysis and redundancy analysis). In these analyses, an independant variable is used as a predictor of the variable(s) of interest (appropriately called dependent variable(s)). In the simplest case, the independent variable is referred to as x and the dependent variable y, and the predictive equation constructed is:
y=b0+b1*x
In more complicated analyses (ie., the multivariate case), there may be multiple independent and dependent variables (matrices X and Y), and the model constructed will consider the interactions not only between, but also within each matrix.