We say that two sets are the same size if you can pair up their elements so that each element of the first set has exactly one partner in the second set and vice versa. It's obvious that this does what we expect for finite sets. For infinite sets, things get less intuitive. There are as many odd numbers as there are even numbers - pair up (1,2), (3,4) and so on. But there are also as many even numbers as there are whole numbers - pair up (2,1), (4,2), (6,3) and so on. There are also as many whole numbers as there are fractions, though the pairing is more fiddly to describe.

So far it looks like there might just be one infinity. But once you include irrational numbers, we find that the set of all real numbers can't be paired up with the whole numbers. Whatever rule you think of to pair up the numbers, I can describe a way to find a real number that you missed out (this proof is called Cantor's diagonal argument). The size of the set of real numbers is a larger infinity than the size of the set of whole numbers.