I am glad you ask and I hope I have sparked your curiosity.

The first question we have to ask ourselves is: What do we mean by infinity? I would answer that somewhat along the lines of "Infinity is something never ending. Something that goes on forever with no end.". The problem with language and infinity is, you quickly get a recursion i. e. you can only describe infinity by using some form of the term infinity (which has a certain irony to it as it will leave you in an infinite loop of definitions).

It is now easy to think that infinite means everything which is certainly not the case. In order to not flood the comment section by a > 1000 words comment I can only give you a short example. The set of all natural numbers is smaller than the set of the real numbers (all numbers, including rational and irrational), although both are infinitely large. As a result there must be different "degrees" of infinity, or whatever term you want to use instead of degrees. The german term for this is "Mächtigkeiten" which I find particularly fitting. It roughly translates to "mightyness".

Maybe part 2 of this series can give you a better understanding of this already.
For a proper example of something larger than the set of natural numbers and proof, you'll have to wait for part 3 though.