Quantile regression allows us to estimate the effect of a set of predictor variables over the entire distribution of the outcome variable or any particular quantile.
Most standard regression techniques focus on estimating how the conditional expectation of an outcome variable ($Y$) depends on a set of predictor variables ($X$). Quantile regression goes beyond mean effects, to estimate the impact of $X$ on any quantile or quantiles of $Y$. This enables researchers to assess many interesting questions like: what is the effect of smoking on infants with the lowest birth weight? How does a job market training program affect those at the bottom of the ability distribution? Or does smaller class size benefit the stronger or weaker students more?