There is an equivalence between hypothesis tests and confidence intervals. (see e.g. http://en.wikipedia.org/wiki/Confidence_interval#Statistical_hypothesis_testing) I'll give a very specific example. Suppose we have sample $x_1, x_2, ..., x_n$ from a normal distribution with mean $\mu$ and variance 1, which we'll write as $\mathcal N(\mu,1)$. Suppose we think that $\mu = m$, and we want to test the null-hypothesis $H0: \mu = m$, at level 0.05. So we make a test statistic, which in this case we will take to be the sample average: $v = (x_1 + x_2 + ... + x_n ) / n$. Now suppose $A(m)$ is the "acceptance region" for $v$ for this test. That means that $A(m)$ is the set of possible values of $v$ for which the null-hypothesis $\mu=m$ is accepted at level 0.05 (I use "accepted" as a shorthand for "not rejected" -- I am not suggesting that you would conclude the null hypothesis is true.). For this example, we can look at the $\mathcal N(m,1)$ normal distribution and choose any set that has probability at least 0.95 under this distribution. Now, a 95% confidence region for $\mu$ is the set of all $m$ for which $v$ is in $A(m)$. In other words, it is the set of all $m$ for which the null-hypothesis would be accepted for the observed $v$. That's why John says "If the CI for an effect does not span 0 then you can reject the null hypothesis." (John is referring to the case of testing $\mu = 0$.)
A related topic is the p-value. The p-value is the smallest level for a test at which we would reject the null-hypothesis. To tie it in with the discussion of confidence intervals, suppose we get a particular sample average $v$, from which we construct confidence intervals of different sizes. Suppose a 95% confidence interval for $\mu$ does not contain $m$. Then we can reject the null-hypothesis $\mu=m$ at level 0.05. Then suppose we grow the confidence interval until it just touches (but doesn't include) the value $m$, and suppose this is a 98% confidence interval. Then the p-value for the hypothesis $\mu=m$ is 0.02 (which we get from 1-0.98).