Salon re-posted a Nautilus piece in which Ayalur Krishnan explains in layman's language what the Continuum Hypothesis is and how its proof can continue to elude us.
A couple of excerpts:
On why the hypothesis can be neither proven nor disproven: "For there to be a proof of the Continuum Hypothesis, it would have to be true in all models of set theory, which it isn’t. Similarly, for the Hypothesis to be disproven, it would have to remain invalid in all models of set theory, which it also isn’t."
On the philosophical ramifications of the unresolved Continuum Hypothesis: "It is natural to wonder whether the uncertainties inherent to mathematics translate into inherent uncertainties about the way the universe functions."