Useful to compare the relative explanatory power of different independent variables. Sometimes this value has real meaning, sometimes not. Generally, when the y-intercept falls within the range of data values, it will be more meaningful than when it falls outside the range of data values. For your models in Assignment 4, your F-scores will tend to be from about 20 to over 200, which are all VERY statistically significant. Beta weights are adjusted partial slopes, or standardized B's.
Beta weights are useful for comparing the relative importance of each independent variable.
The Adjusted R-Square takes into account not only how much of the variation is explained, but also the impact of the degrees of freedom. That is, look at the adjusted R- Square to see how adding another variable to the model both increases the explained variance but also lowers the degrees of freedom. This serves as a disincentive to simply throwing in a huge number of variables into the model to increase the R-square. In regression the goal is to find the best fitting equation that links the independent variables with the dependent variable. This is one that minimizes the error of prediction.