Determine wether a set of sets contains a partition of a given set
Without loss of any really interesting aspect, we can assume that $S_i\subseteq G$ for all $i\in I$. Otherwise, we can we can remove each $S_i$ which is not a subset of $G$. This reduction step takes $O(ng)$ times, where $n=\sum_{i\in I}|S_i|$ and $g=|G|$.

Now that $S_i\subseteq G$, this problem is the exact cover problem, which is one of Karp's 21 NP-complete problems. You can find all sort of information about it at its Wikipedia entry.

I have played briefly a game called Pentomino, which is an exact cover problem.