What is the difference between the R-Squared and adjusted R-Squared when running a regression analysis?

Adjusted R-squared applies only to multiple regression

As you add more independent variables to a multiple regression, the value of R-squared increases giving you the impression that you have a better model which isn't necessarily the case. Without going in depth, the adjusted R-squared will take into account this bias of increasing R-squared.

If you examine any multiple regression results, you will note that the adjusted R-squared is ALWAYS less than R-squared because the bias has been removed.

The goal of the statistician is to optimize the best combination of independent variables such that the value of adjusted R-squared is maximized.

hope that helps