This is something that bothers me for quite some time, but I didn't find yet a satisfactory answer. I hope that the wisdom of the people hear will help me to clarify this:
In a multivariate regression, right-side variables can be included to control for the contribution of the other variables. For example, in a regression that predicts, let's say, math abilities by shoe-size, we might get a strong relationship (because younger children tend to have smaller feet and less mathematical knowledge). But when including the covariate "age" in this regresion, the relationship between math abilities and shoe-size would disappear.
Thus, the regression estimates provide the partial correlation of the predictor with the dependent variable (that is, the estimate captures the contribution of that predictor beyond the contribution of all the other predictors).
However, by including the control variable of "age" in our regression - aren't we creating a problem of collinearity? As age is correlated with shoe-size (otherwise, why would we want to control for it?). Isn't that a problem for obtaining accurate estimates?