Endogeneity refers to a situation where an explanatory variable in a model is correlated with the error term. Endogeneity induces biased parameter estimates. This is an important problem when working with observational data and the goal is causal inference.
If two variables, $X_1~\&~X_2$, are correlated with each other and both are entered into a model, the result will be some amount of collinearity, if one is left out of the model, and it actually does have an effect on the response, the result will be endogeneity. The effect of the omitted variable will be attributed to the variable that is in the model by virtue of their correlation, causing bias. Endogeneity can also arise from other sources, such as measurement error in $X$.
When working with observational data excluding the possibility of endogeneity is rarely credible. Thus, it is a substantial barrier to drawing causal conclusions (although it is not a problem for making predictions or assessing marginal relationships).
A number of techniques (other than experiments) have been developed to help address this problem. They include instrumental-variables, propensity-scores, regression discontinuity designs, difference in differences (a.k.a., did), quasi-experimental designs, natural experiments, etc.
