What is set of measure zero?

From an application point of view, a set of measure zero has the property that you can change the value of the function at points in the set without affecting the value of the integral of the function.

A set of measure zero, at least in terms of Lebesgue measure (as your tags suggest), is simply a set that's so small that we can contain it in countably many open balls with total volume being arbitrarily small.