It's really not a complicated idea. There are sets. For example, the set of natural numbers or the set of rational numbers. Turns out there is a way to pair up 1-1 elements of the set of natural numbers and the set of rationals. Both are infinite. You might think, perhaps you can pair up the elements of any two infinite sets? You can pair up the naturals with the even numbers, the naturals with the rationals, the rationals with the algebraic numbers.
Turns out there are infinite sets of numbers that cannot be paired up with the naturals. One example is the set of subsets of natural numbers, called P(N). x is an element of P(N) if x is a subset of natural numbers.
I will prove it by contradiction. Assume there was a way to pair up P(N) with N in a 1-1 way. For every natural number n, let f(n) be the corresponding subset of N in the pairing. Then we can define a subset A of the naturals by saying that A= set of all naturals n such that n does not belong to the set f(n).
A is a subset of the naturals and so there is some natural number k such that f(k)=A. Does k belong to A? If it did, by definition of A k would not belong to f(k), which is A, so we get a contradiction. If it did not belong to A, then it would satisfy "k does not belong to f(k)" hence would be a member of A, also a contradiction. We see that in any case we get a contradictory result, which means that our assumption that such a pairing f exists is false.
However, there is a way to realize the naturals as a subset of P(N), namely by the correspondence n->{n}, so in that sense N is not larger than P(N).
Thus we see there are infinite sets that are intuitively larger than others. If we define an infinity by an equivalence class of infinite sets that can be paired up 1-1, (there are some technical difficulties with this definition but the axiom of regularity takes care of them), we see that in fact there are some larger infinities than others.
