>>13296448if you want to understand how it's done, watch Vsauce like
>>13296460 suggests
if you want a perspective on why it's important, i'll give you a quick rundown. in analysis you have the "measure problem", that is assigning a meaningful notion of length/area/volume to sets in some space. this idea originated with Lebesgue, which led to a revolution in analysis and the creation of measure theory. what one does is try and construct a function that takes as its input subsets of some set and produces a nonnegative real number (possibly infinity) that is meant to represent the subset's "volume". this function should have at least some of the properties we typically expect length/area/volume to have: e.g. it should be invariant under rigid motions (if you move around a line segment or rotate it or reflect it, its area won't change), it should be additive (if you can break your set up into non-overlapping pieces, the measure of the entire set should be equal to the sum of measures of its constituent parts) and so on. now it turns out that that in its full generality, the measure problem is impossible to solve in R^n (might be you can do it in different spaces, idk): there is no way of assigning a reasonable notion of measure to every possible subset of R^n, because there might exist extremely pathological ones. so what mathematicians do is content themselves with only measuring certain well-behave subsets of R^n, the so-called "measurable" ones. iirc, usually they're defined as something that can be approximated by boxes/rectangles
the problem with the patological examples, is that in order to construct them one needs the Axiom of Choice. that is, at some step in the construction of such sets, there comes a point where you have to make an infinite amount of choices and that is in general problematic. i only know of Banach-Tarski and the Vitali set, so there might exist non-measurable sets that can be constructed without the AoC, but i doubt it
