What is a weak value really? There have been a lot of recent experiments performing weak measurements. Some of the conclusions seem to be quite surprising (e.g. this paper) and there is still debate if the weak measurement is actually a "measurement" or not. My question is what the weak value really represents. I know what it is mathematically (see e.g. wikipedia), but what is it physically? Is it an actual property of the system or an inferred quantity? And why is it useful to measure it (as in what does it tell us up the system)?
 A: Take a large ensemble of particles and preselect state $\left|\psi\right\rangle$
Then take a large sub-ensemble of this ensemble and postselect state $\left|\phi\right\rangle$.
In other words, set up the experiment, with the ensemble starting in state $\left|\psi\right\rangle$. Make a weak measurement of the observable $\hat{A}$, then make a strong measurement (at the end of the experiment) and look at the data that ended in state $\left|\phi\right\rangle$. As mentioned in the comments of the question, this does potentially lead to issues about whether or not this is actually a weak measurement as it involves a strong measurement of the measurement device.
The weak value, $A_W$, gives the average (mean) value of $\hat{A}$ that the particles in this sub-ensemble appear to have when measured via a weak interaction
It is not relevant or useful when looking at a single particle, although measuring an ensemble of particles is equivalent to repeating an experiment with a single particle, many times.
If, instead of a weak measurement, a strong measurement was performed, then the wavefunction collapses and we no longer know which particles would have ended in state $\left|\phi\right\rangle$ given that they started in state $\left|\psi\right\rangle$ as they have effectively now started in a different state.
Postselecting $\left|\phi\right\rangle$ such that $\left\langle\phi|\psi\right\rangle$ is small allows for amplification of measurements that were previously too small to observe (e.g. Spin Hall Effect of Light - http://science.sciencemag.org/content/348/6242/1448)
They also bring a new way to look at counterfactual statements - a set of statements that classically, cannot be simultaneously true as they appear to contradict each other. This is usually solved in quantum physics by saying that when making a (strong) measurement of such a statement, the wavefunction collapses and the other statements cannot be measured so do not have to be simultaneously true. As weak measurements do not collapse the wavefunction, they allow for these statements to be tested (e.g. Hardy's Paradox - http://www.tau.ac.il/~yakir/yahp/yh12)
