For the edge ideal $I$ of an arbitrary simple graph $\Gamma$ we describe the monomials of the saturation of $I^t$ in terms of (vertex) weighted graphs associated with the monomials.  This description allows us to characterize the embedded associated primes of $I^t$ as covers of $\Gamma$ which contain certain types of subgraphs of $\Gamma$. As an application, we completely classify the associated primes of $I^2, I^3$ and $I^4$  terms of $\Gamma$.