Regression analysis is a statistical technique that looks for correlations between a range of variables in research structures - it might look for correlations between test scores and pupils' ages or whether they have free school meals. The technique is often used to suggest that one factor is not just correlated with a final result, but causes it. This is difficult to prove, as it is not always easy to find out what it is in a factor that might cause the result, but it makes a convenient shortcut for people who want to argue that one factor - often income - causes another - often educational achievement,.
So, we can all be grateful to Professor Steven Gorard for this note in his chapter on Multiple Linear Regression in Arthur, J., Waring, M., Coe, R. & Hedges, L.V. (2012). Research Methods and Methodologies in Education.
A zero or weak correlation coefficient does not mean that there is no interrelationship between the variables involved. It might just mean that the relationship is more complex than a single linear correlation. Also, regression does not test anything. It merely models a relationship that we predict as analysts. … It is certainly not evidence of a specific, or indeed any, causal relationship between the variables involved. p. 353
This will, of course, not prevent misuse of the technique for political purposes, but at least it makes the position clear.