In layman's terms, what is a quantum fluctuation? What causes it and how does it occur?  If you do post some mathematics, please explain what each term means too please.
 A: Quantum fluctuations are a popular buzzword for the statistical 
triviality that the variance (the spread of values)  of a random variable A (in context of quantum physics, this could be the position of a particle or the amount of energy that it has) with zero mean is typically not zero - except that A is now an operator. Some people, therefore, think that this deserves a much more mysterious name. 
Taken from the section ''Does the vacuum fluctuate?'' in 
Chapter A8: Virtual particles and vacuum fluctuations of

A theoretical physics FAQ
A: Fluctuations in the mean are also called fluctuations. It gives a notion about how reliable the mean value is (the second moment of the distribution). Any quantity that we are uncertain about will have that uncertainty encoded in a probability distribution, Quantum mechanics is no different in that respect then any other theory of inference, it is only different in that it claims that the uncertainty is intrinsic whereas other theories of inference simply assume that the data is observable in principle but not in practice.
We use the term `quantum fluctuation' therefore to impose the idea of fluctuations on physical variables that we classically thought of as being exact and obtainable such as position and momentum.
An interesting and quick calculation in scalar free field theory gives an interesting example of `quantum fluctuations'
$$\langle \phi(x)\rangle_0=0$$
$$Var(\phi)_0=\langle\phi(x)^2\rangle_0-\langle \phi(x)\rangle_0=\langle\phi(x)^2\rangle_0
=\int \frac{d^3k}{(2\pi)^3}\frac{1}{\sqrt{\vec k^2+m^2}}\rightarrow\infty$$
The average value of the field is vanishing, but when we ask the extent to which this result can be trusted, it cannot, our ignorance is infinite.
