Cantor's diagonal argument: why the reals cannot be listed
Sizes of infinity: a strict hierarchy
The natural numbers, integers, and rationals are all countably infinite: they can all be put in a one-to-one correspondence with each other. The real numbers are uncountably infinite: a strictly larger infinity. Between these two sizes, the Continuum Hypothesis asks whether there is anything in between.
Hilbert's Hotel: a hotel with infinitely many rooms, all full, always has room