Absolute minimum is where, in the infinite domain of f(x), the least value the function is the absolute minimum.
However, the local minimum is only a specific interval on the domain of f(x) where the lowest value is, but only in that interval.
So, usually, this comes up when you're dealing with 1st and 2nd derivatives. The first derivative tells you where local max and mins are, where the 2nd derivative tells you if the concavity is up or down.
