Gödel's Incompleteness Theorem

Hi,

I was wondering if someone could please provide an example of something that is "unprovable"? Is the Continuum Hypothesis such an example?

Also, I remember reading a while ago about the possible existence of theorems that were "unprovable" and so were true (because if they were false there would exist a counter-example, which would be a proof that they were false). However, surely this is a contradiction? Surely any example that could be proven false by a counter example cannot be proven to be unprovable?

I hope those questions make sense...