Correlation is a measure of linear association between two variables X and Y, while linear regression is a technique to make predictions, using the following model:
Y = a1 X1 + ... + ak Xk + Error
Here Y is the response (what we want to predict, for instance revenue) while the Xi's are the predictors (say gender, with 0 = male, 1 = female, education level, age, etc.)
Typically, the predictors are somewhat correlated to the response. In regression, we want to minimize the absolute value of the correlation between the observed response and the error term. We choose the parameters a1, ..., ak that accomplish this goal.
There are various types of correlation coefficient as well as regression. For more details, see