Why is the Jeffreys prior useful? I understand that the Jeffreys prior is invariant under re-parameterization. However, what I don't understand is why this property is desired.
Why wouldn't you want the prior to change under a change of variables?
 A: While often of interest, if only for setting a reference prior against which to gauge other priors, Jeffreys priors may be completely useless as for instance when they lead to improper posteriors: this is for instance the case with the simple two-component Gaussian mixture
$$p\mathcal{N}(\mu_0,\sigma_0^2)+(1-p)\mathcal{N}(\mu_1,\sigma_1^2)$$
with all parameters unknown. In this case, the posterior of the Jeffreys prior does not exist, no matter how many observations are available. (The proof is available in a recent paper I wrote with Clara Grazian.)
A: Suppose that you and a friend are analyzing the same set of data using a normal model. You adopt the usual parameterization of the normal model using the mean and the variance as parameters, but your friend prefers to parameterize the normal model with the coefficient of variation and the precision as parameters (which is perfectly "legal"). If both of you use Jeffreys' priors, your posterior distribution will be your friend's posterior distribution properly transformed (don't forget that Jacobian) from his parameterization to yours. It is in this sense that the Jeffreys' prior is "invariant"
(By the way, "invariant" is a horrible word; what we really mean is that it is "covariant" in the same sense of tensor calculus/differential geometry, but, of course, this term already has a well established probabilistic meaning, so we can't use it.)
Why is this consistency property desired? Because, if Jeffreys' prior has any chance of representing ignorance about the value of the parameters in an absolute sense (actually, it doesn't, but for other reasons not related to "invariance"), and not ignorance relatively to a particular parameterization of the model, it must be the case that, no matter which parameterizations we arbitrarily choose to start with, our posteriors should "match" after transformation.
Jeffreys himself violated this "invariance" property routinely when constructing his priors.
A: Let me complete Zen's answer. I don't very like the notion of "representing ignorance". The important thing is not the Jeffreys prior but the Jeffreys posterior. This posterior aims to reflect as best as possible the information about the parameters brought by the data. The invariance property is naturally required for the two following points. Consider for instance the binomial model with unknown proportion parameter $\theta$ and odds parameter $\psi=\frac{\theta}{1-\theta}$.


*

*The Jeffreys posterior on $\theta$ reflects as best as possible the information about $\theta$ brought by the data. There is a one-to-one correspondence between $\theta$ and $\psi$. Then, transforming the Jeffreys posterior on $\theta$ into a posterior on $\psi$ (via the usual change-of-variables formula) should yield a distribution reflecting as best as possible the information about $\psi$. Thus this distribution should be the Jeffreys posterior about $\psi$. This is the invariance property.

*An important point when drawing conclusions of a statistical analysis is scientific communication. Imagine you give the Jeffreys posterior on $\theta$ to a scientific colleague. But he/she is interested in $\psi$ rather than $\theta$. Then this is not a problem with the invariance property: he/she just has to apply the change-of-variables formula.
A: To add some quotations to Zen's great answer: According to Jaynes, the Jeffreys prior is an example of the principle of transformation groups, which results from the principle of indifference:

The essence of the principle is just: (1) we recognize that a
  probability assignment is a means of describing a certain state i
  knowledge. (2) If the available evidence gives us no reason to
  consider proposition $A_1$ either more or less likely than $A_2$, then
  the only honest-way we can describe that state of knowledge is to
  assign them equal probabilities: $p_1=p_2$. Any other procedure would
  be inconsistent in the sense that, by a mere interchange of the
  labels $(1, 2)$ we could then generate a new problem in which our
  state of knowledge is the same but in which we are assigning different
  probabilities…

Now, to answer your question: “Why wouldn't you want the prior to change under a change of variables?”
According to Jaynes, the parametrization is another kind of arbitrary label, and one should not be able to “by a mere interchange of the labels generate a new problem in which our state of knowledge is the same but in which we are assigning different probabilities.”
A: Jeffreys prior is useless. This is because:


*

*It just specifies the form of the distribution; it does not tell you what its parameters should be.

*You are never completely ignorant - there is always something about the parameter that you know (e.g. often it cannot be infinity). Use it for your inference by defining a prior distribution. Don't lie to yourself by saying that you don't know anything.

*"Invariance under transformation" is not a desirable property. Your likelihood changes under transformation (e.g. by the Jacobian). This does not create "new problems," pace Jaynes. Why shouldn't the prior be treated the same?


Just don't use it.
