lightcastle | Greater and Lesser Infinities [Ask the Interweb Brain]

A question for my math-geek readers: are there infinities of differing sizes? I remember being quite certain there were at some point, but I am quite sure I am mistaken as to why. (I remember someone telling me that despite the intuitive obviousness of 0-infinity being smaller than 1-infinity, the truth is those are the same size.)

So, without necessarily trying to explain why in great detail, can someone confirm whether there are infinities of different sizes?