A local extremum is a local maximum or a local minimum.
For a graph (a function of one parameter), y=f(x), a local extremum is a critical point (a point where the first derivative of the function is zero), where the sign of the derivative is different on the left and the right (making it the first derivative equivalent of an inflection point).
For a function of more than one variable, y=f(x1, … xn) a local extremum occurs where all of the first derivatives (∂y/∂x1, … ∂y/∂xn) are zero, and the curvature is positive.
C-Dawg has reminded me that a function may have a local extremum at a point which is not differentiable. Generally, such instances must be handled on a case-by-case basis. Fortunately, it usually easy to recognise where a function is not differentiable - but to find out if any of those points are actually local extrema, well, that's more tricky.
tdent indicates that a critical point can be classified as a local extremum if the first nonzero derivative is an even derivative (second derivative, fourth derivative, etc.), and it is not a local extremum if the first nonzero derivative is an odd one (third derivative, or fifth, etc.).