What does the notation $g\colon X \to Y$ mean in this Wikipedia page, under the section "Problem statement (supervised version)"?
The $g$ function there is the mapping between instances to output labels. In other words, it is the correct association of events and their respective patterns. Presumably, the goal of supervised pattern recognition is to assign patterns to events in a way that best mirrors the truth (i.e. the $g$ function).
It describes a function; a mapping from one set (X) to another (Y). For a more details see Set-builder notation