The y-intercept of a graph (or function, y = f(x)) is where the graph intercepts the y-axis.
The y-axis is given by the equation x = 0, so the y-intercept of a function is the solution to the simultaneous equations y = f(x) and x = 0, or, x = 0 and y = f(0) - also written (0, f(0)).
It is possible that a function is undefined at zero (e.g. f(x) = 1 / x), in which case there is no y-intercept; if the function is defined at zero then there is one intercept. There can never be more than one.
C-Dawg has (correctly) pointed out to me that a relation (which frequently gets confused for a function) can have more than one y-intercept, and that parametric functions sometimes used for describing graphs (e.g. x(θ) = cos(θ), y(θ) = sin(θ)) can have more than one y-intercept too. In all cases, to find the y-intercept(s), solve whatever equations are given with the constraint x = 0.