To elaborate on what Liz said, the way you compare sizes of infinite sets is by constructing functions between them. If you can place the items in the two sets in a one-to-one correspondence, the sets are the same size. If you try doing that between set A and set B, but there are elements in set A with no corresponding element in set B, then set A is bigger.

Cantor's diagonalization argument proves that there are more reals than integers. I found some explanations that aren't too jargony here. It's a great introduction to reasoning about infinities. If you're interested in the topic in general, I highly recommend Everything and More by David Foster Wallace, regardless of your level of math knowledge.