What kind of information is Fisher information? Suppose we have a random variable $X \sim f(x|\theta)$. If $\theta_0$ were the true parameter, the the likelihood function should be maximized and the derivative equal to zero. This is the basic principle behind the maximum likelihood estimator. 
As I understand it, Fisher information is defined as 
$$I(\theta) = \Bbb E_\theta \Bigg[\left(\frac{\partial}{\partial \theta}f(X|\theta)\right)^2\Bigg ]$$
Thus, if $\theta_0$ is the true parameter, $I(\theta) = 0$. But if it $\theta_0$ is not the true parameter, then we will have a larger amount of Fisher information. 
In what sense is Fisher information "information"? Does it measure the "error" of our likelihood estimator in maximizing the true parameter? Or is that the wrong way to think about it? How is this different from the way Shannon defines information? 
 A: I think this might be more appropriate for cross-validated. Nevertheless here is one way you can look at it. Let $f(x,\theta)$ be a probability distribution with unknown parameter $\theta$, let $\hat{\theta}$ be an unbiased estimator of $\theta$. Then we have the well-known Cramer-Rao lower bound:
$$
E_{\theta}(\hat{\theta}-\theta)^{2}\ge \frac{1}{I(\theta)}
$$
In other words the more information we have measured by $I(\theta)$, the more likely we can obtain a better unbiased estimator for $\theta$. This is related to the concept of asymptotic efficiency proposed by Lehman in 1970s. In folklore language among the statistical community, it is often stated as "you cannot beat the $\sqrt{n}$ speed (coming from central limit theorem)". And it played a paramount role in the field of non-parametric statistics. So fisher information is closely-connected to many different subfields in statistics. You can easily find a lot of material related to it by Googling online. 
Added:
It might be worth while to point out that $I(\theta')$ is almost never zero even if $\theta'=\theta$ the real parameter well-approximated by the MLE. To see this we have
$$
I(\theta')=\int_{D} (\partial_{\theta}\log(f(x,\theta'))^2f(x,\theta')dx
$$
Notice the integral is over different $x$ values in the domain $D$. For any given $x$, we can try to find the MLE $\hat{\theta}$ via (does not always work)
$$
\frac{d}{d\theta}L(x,\theta')|_{\hat{\theta}}=0, \forall x\in D
$$
But the trouble is in the above you need to force all the $\partial_{\theta}\log(f(x,\theta))^2$ to be zero, which is almost impossible for fixed $\hat{\theta}$ unless $D$ is of point measure. So in general the fisher information is never zero even for $\theta=\hat{\theta}$ be the MLE for some given $x$- it measures the sensitivity of the overall $f(x,\theta)$ on $\theta$ instead. The larger it is, the more sensitivity you have, and the better chance for a nicer unbiased-estimator of $\theta$. 
Added:
I realized one way to think about the question when $n\rightarrow \infty$ is the score test, which is the most powerful test at small deviations. Not sure if this helps. For me it is a different way of interpreting the original question. 
