let's call Runge an open subset $\Omega \subset \mathbb C $ such that polynomials are dense in $\mathcal H(\Omega)$ . A hole of $\Omega$ is a compact connected component of $ \mathbb C \setminus \Omega $. We then have the equivalent statements, for the open subset $\Omega \subset \mathbb C$ (not assumed connected).
a) $\Omega$ is Runge
b) $\Omega$ has no hole
c) Every connected component of $\Omega$ is simply connected
In the general case, when these equivalent conditions are not fulfilled, Runge's theorem says that if you choose one point in each hole of $\Omega$, then the rational functions with poles only in these points are dense in $\mathcal H(\Omega)$. Beware that you can have a non-denumerable set of holes : take for $\Omega$ the complement of a Cantor set in $\mathbb R \subset \mathbb C$.
In a related vein, Mergelyan's (difficult) theorem says that if you take a compact subset $K \subset \mathcal H (\Omega)$ with connected complement in $\mathbb C$, then any continuous function on $K$ which is holomorphic in the interior of $K$ can be uniformly approximated by polynomials.
Bibliography: Remmert's book is probably the best reference for this question ( and many others...) The equivalence of the statements quoted above is proved in Chapter 13, section 2.
Mergelyan's theorem is not in Remmert's book but is proved on page 386 of Rudin's well known Real and complex analysis, of which you can find a review here
Remark: These results are somewhat astonishing. Take $\Omega=\mathbb C \setminus \{x\in \mathbb R| x \leq -1\}$ and for $f \;$ the holomorphic branch of the logarithm $f(z)=log (1+z) $ which is zero at the origin. Its Taylor series $\sum_{k=0}^{\infty} (-1)^k \frac {z^k}{k}$ diverges for $|z|\gt 1$ and the partial sums of the series are polynomials which definitely don't converge to $f$. However, since $\Omega$ has no holes, there does exist some sequence of polynomials converging to $f$ uniformly on compact subsets of $\Omega$.
