Non-significant in correlation, but significant predictor in regression. How to explain suppression? I'm having trouble explaining some results...
I have 5 independent variables (A, B, C, D and E) and I want to know their relation to the dependent (Y). 
Only variables A and C are significantly positively correlated with Y.
Then, in a multiple regression (model was significant with a high F) only variables D and E were found to be significant predictors.
I’ve read a little bit about suppression and tolerance (tolerance statistics are fine in my case), but I’m unsure how to explain what is happening here. 
Do I explain that variables A and C are suppressed in the regression model? Or do I explain that variables D and E were suppressed in the correlations?
For the research question (predictors of Narcissism), in my opinion it would make more sense that higher scores on A (perceived unfairness of childhood discipline) and C (reward orientation) are more closely related to Y (Narcissism) than D (loneliness) and E (feeling socially supported by friends and family).
I’ve read similar questions here and elsewhere but cannot find a specific answer about how to explain where and on which variables suppression occurs.
 A: Whenever independent variables(IVs) are not perfectly orthogonal, this sort of thing can happen.
In observational studies, and certainly in psychology and similar fields, IVs are almost always far from orthogonal. At the extreme, there is perfect collinearity. Slightly less extreme is collinearity that makes all estimation really hard. You don't have either of those - that's what the reasonable VIFs are telling you - but still, your IVs are not orthogonal. 
In your case you have:
A - perceived unfairness of discipline - sig bivariately but not multi
B - ???  - not sig at all
C - Reward orientation - sig bivariately but not multi
D - Loneliness  - Sig multi but not bi
E - Social support  - Sig multi but not bi
First, I suggest looking at the effect sizes and not at significance. Certainly not a dichotomized "sig/not sig".
Second, it seems you have suppression and mediation going on here (although people define both those terms in slightly different ways). Rather than get hung up on definitions, let's think about what's going on.
A and C lose something when controlling for B, D and E.  That is, A becomes lower when you are looking at people with similar scores on B, C, D and E and C becomes lower when looking at people with similar scores on A, C, D and E.  On the other hand, D and E gain something. That is, D becomes higher (a better predictor) when controlling for A, B, C and E and similarly for E.
Frankly, those results surprise me. But that's what your results are saying. 
