What's a "p-value"?

Many resources provide informal definitions. There seem to be two equivalent ones:

1) The smallest level at which would be rejected.

2) The probability that you would obtain evidence against which is at least as strong as that actually observed, if were true.

I'm looking for a formal statement of both of these definitions (preferably resorting to little or no measure theory), as well as for a proof of their equivalence.

Thanks.