What's a "p-value"?

Many resources provide informal definitions.

There seem to be two equivalent ones:

1) The smallest level at which $\displaystyle H_0$ would be rejected.

2) The probability that you would obtain evidence against $\displaystyle H_0$ which is at least as strong as that actually observed, if $\displaystyle H_0$ 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.