The always-excellent Vi Hart (previously) gives one of the most engaging explanations of multiple infinities we've come across in some time.
This problem of infinity was pondered by Georg Cantor. What he concluded started him down a road that wound through infamy, through respectability, and wound up in theology. Find out more than anyone ever cared to consider about the infinite.
The notion of infinity is fundamentally beyond the human ability to comprehend, but that hasn't stopped mathematicians from trying. So just what is infinity, and why is there more than one of them? And just what is infinity plus one?