A locally constant function is a function whose value never changes. This is a weaker concept than that of a constant function, which has only one value. They correspond on a connected space. However, in general, a function may be locally constant but not constant, since it can take values on two distant components without the values' ever changing between them (since there is no path between them).
A locally constant function is a section of a constant sheaf;
a locally constant stack is a section of (… and so on…)
A locally constant sheaf / -stack is also called a local system.