the real numbers is the set that includes the irrational, rationals, integers, whole numbers and natural numbers. that is, all those sets are subsets of the reals.

so the real numbers are made up of rational and irrational numbers. the irrationals are by themselves. the rationals though include other sets. the integers are a subset of the irrationals, the whole numbers are a subset of the integers (they are pretty much the same thing. i guess we wouldn't call 0 a whole number which is the reason for the distinction), the natural numbers are a subset of the integers (natural numbers are positive integers). so that's it. the diagram is just showing how the different sets of numbers relate to each other, i.e. which set contains which