A Vitali set is a specific type of set in mathematics that demonstrates the concept of non-measurable sets. It is constructed using the properties of Lebesgue measure, which is a way to assign a size or measure to subsets of Euclidean space. The Vitali set is created by selecting points from the interval [0, 1] in such a way that no two points differ by a rational number. The significance of the Vitali set lies in its ability to show that not all subsets of real numbers can be measured in the traditional sense. This challenges the intuitive notion of size and leads to deeper discussions in set theory and real analysis.