Partial correlation to check whether C drives a normal correlation between A and B Say I have two scores from a certain population, A and B, that are correlated. Say that I also have a third variable from that population, C, and I now want to see if the partial correlation between A and B still holds when C is added as a covariate.
If the correlation no longer holds, is it correct to conclude that C was "driving the effect", i.e. it explains enough variance in one variable such that there isn't much left for the other variable? 
If so, can I also conclude that C is correlated with A and/or B? 
And is it in fact correct to assume trasitivity of the correlation relationship, i.e. if A correlates with B, and if this correlation no longer holds when controlling for C, then it must mean that C correlates with both A and B?
 A: 
Can I also conclude that C is correlated with A and/or B?

Yes, for if not, then controlling for $C$ would have not changed the correlation between $A$ and $B$.

If A correlates with B, and if this correlation no longer holds when controlling for C, then it must mean that C correlates with both A and B?

Yes.
We can relate the correlation coefficients to one another.  To make this easier we may as well assume, with no loss of generality, that $A$, $B$, and $C$ have all been standardized, because those operations do not change the correlations or partial correlations.  The covariances among the three variables are therefore their correlations $\rho_{AB}, \rho_{BC},$ and $\rho_{AC}$.
Let $A_C$ and $B_C$ be the residuals after regressing $A$ and $B$ on $C$.  The regression equations are
$$A = \rho_{AC}C + A_C,\ B = \rho_{BC}C + B_C$$
"This correlation no longer holds" means $A_C$ and $B_C$ are uncorrelated, implying their covariance is zero.  Use this fact, and the lack of correlation between $C$ and any of the residuals $A_C$ or $B_C$, to compute the correlation
$$\eqalign{
\rho_{AB} &= \text{Cov}(\rho_{AC} C, \rho_{BC} C) +  \text{Cov}(\rho_{AC} C, B_C) + \text{Cov}(A_C, \rho_{BC} C) + \text{Cov}(A_C, B_C) \\
&=\rho_{AC}\rho_{BC} + \rho_{AC}(0) + \rho_{BC}(0) + 0 \\
&= \rho_{AC}\rho_{BC}.
}$$
It is now clear that $\rho_{AB} \ne 0$ if and only if both $\rho_{AC}\ne 0$ and $\rho_{BC} \ne 0$, QED.
