Importance sampling is a variance reduction technique to approximate integrals/expectations which are not directly computable.
Importance sampling is a method to evaluate an integral of the form $$E[h(X)] = \int h(x)f(x) dx$$ when this cannot be directly computed. Instead of sampling from the density $f$ as in classical Monte Carlo integration, importance sampling (IS) samples from an arbitrary density $g$ which must be a valid pdf: $$E[h(X)] = \int h(x)\frac{f(x)}{g(x)}g(x) dx$$ Based on samples $X_1,...,X_n$ generated from $g$ (and not from $f$), the expectation under $g$ converges in probability to $$\frac{1}{n} \sum_{j=1}^{n} \frac{f(X_j)}{g(X_j)}h(X_j) \rightarrow E[h(X)]$$ as in classical Monte Carlo integration. IS has the advantage that it puts very little restrictions on the blanket function $g$. Usual choices for $g$ are standard distributions that are either easy to simulate or which are efficient in the approximation of the integral. Directly sampling from $f$ as in Monte Carlo integration is normally not efficient hence IS sampling requires fewer samples to achieve the desired result.
For further reference see Robert, C.P. and Casella, G. (2004) "Monte Carlo Statistical Methods", 2nd Edition, Springer Texts in Statistics, Springer Science+Business Media Inc., New York
