A previous node suggested the following:
A statistical method in which the relationship between the mean value of a random variable and the corresponding values of one or more independent variables is observed.
I would like to clarify a few things. Only in the most technical sense is this correct, but it fails to note many of the subtleties, some of which I discuss below.
1) The node above is only technically correct for what we call Ordinary Least Squares Regression, or OLS which has a variety of assumptions associated with it. Among these assumptions is that the independent and dependent variables are normally distributed, random variables. However, this assumption can be loosened, thus making the general definition above actually too general.
2) Regression is almost always employed when there are MORE THAN one independent variables. When we have simply one dependent and one independent variable, the Beta coefficient that is generated is functionally equivalent to r, or the correlation coefficient. Furthermore, bivariate ( one independent one dependent variable) almost certainly violates another one of the assumptions of regression, that of specification. An ill specified model (regression of course being a model of reality) leads to estimators ( in the form of Beta coefficients) which are unreliable. Sadly, there is no way of knowing a priori if they are biased upwards or downwards.
There are additional considerations to take into account with regression, but these should go into another discussion.