In ordinary regression, we are interested in modeling the mean of a continuous dependent variable as a linear function of one or more independent variables. This is often what we do, in fact, want, and this form of regression is extremely common.
Sometimes, though, we want something else. Sometimes the dependent variable isn’t continuous and we turn to logistic regression or some form of count regression. Sometimes the dependent variable is censored, as a time to event, and we turn to survival analysis.
But sometimes even though the dependent variable is continuous, we are not interested in the mean, but in some other statistic about the population. One such situation is when we want to model some quantile (also known as percentile) of the population. That is, we might be interested not in what affects the mean, but in what affects (say) the 3rd quartile, or the 95th percentile, or some other percentile.
When might we want this?
Suppose our dependent variable is bimodal or multimodal – that is, it has multiple “humps”. If we knew what caused the bimodality, we could separate on that variable and do stratified analysis, but if we don’t know that, quantile regression might be good.
If our DV is highly skewed – as, for example, income is in many countries – we might be interested in what predicts the median (which is the 50th percentile) or some other quantile.
One more example is where our substantive interest is in people at the highest or lowest quantiles. For example, if studying the spread of sexually transmitted diseases, we might record number of sexual partners that a person had in a given time period. And we might be most interested in what predicts people with a great many partners, since they will be key parts of spreading the disease.