Causal inference tries to quantify the effect of a change in $X$ on $Y$ while holding constant or eliminating all other relevant factors which might influence this relationship. Such an effect is often referred to as a causal (or treatment) effect. It can be used to answer "what if"-type questions like: what happens to crime if we increase the police force by 10%? How will a person's future earnings change if she goes to school one year longer?
Note that in statistics and philosophy there is no single agreed-on definition of causation. Commonly used conceptual frameworks to think about causality are the Rubin Causal Model based on counterfactuals, structural equation modelling, and causal decision theory. Statistical tools for causal inference include instrumental variables models, regression discontinuity designs, difference-in-differences, and matching models.
For further references on the philosophical and statistical aspects of causal inference see: