If two variables have 0 correlation, why are they not necessarily independent? Are zero correlated variables independent under special circumstances ?  If possible, I am looking for an intuitive explanation, not a highly technical one.
Correlation measures linear association between two given variables and it has no obligation to detect any other form of association else.

So those two variables might be associated in several other non-linear ways and correlation could not distinguish from independent case.

As a very didactic, artificial and non realistic example, one can consider $X$  such that $P(X=x)=1/3$ for $x=-1, 0, 1$ and $Y=X^2$. Notice that they are not only associated, but one is a function of the other. Nonetheless, their correlation is 0, for their association is orthogonal to the association that correlation can detect. 
