Instrumental variables (IV) are used for causal inference with observational data in the presence of endogeneity when standard regression methods yield biased and inconsistent estimates.
Instrumental variables (IV) are used for causal inference with observational data in the presence of endogeneity (e.g. due to omitted variables or reverse causality) when standard regression methods would yield biased and inconsistent estimates. It is possible to consistently estimate the parameter of interest if there exists a variable $z$ (called an "instrument") which is highly correlated with the endogeneous variable $x$ but uncorrelated with the error term $u$ and which affects the outcome $y$ only through $x$:
$$\begin{matrix}
z & \rightarrow & x & \rightarrow & y \newline
& & \uparrow & \nearrow & \newline
& & u &
\end{matrix}$$
Instrumental variables models like the standard IV estimator or 2-stage least squares utilize the exogenous variation in $z$ to separate the effect of $x$ on $y$ from the unwanted influence due the correlation between $x$ and $u$. The difficulty in applied work is to find good instruments. Weak correlation between $z$ and $x$ leads to inconsistent estimation as do even small correlations between $z$ and $u$. Natural experiments or randomized trials are the usual sources for good instruments.
For an overview and a list of examples see Angrist and Krueger (2001).
