KL divergence KL(p||q), in the context of information theory, measures the amount of extra bits (nats) that is necessary to describe samples from the distribution p with coding based on q instead of p itself. From the Kraft-Macmillan theorem, we know that the coding scheme for one value out of a set X can be represented q(x) = 2^(-l_i) as over X, where l_i is the length of the code for x_i in bits.

We know that KL divergence is also the relative entropy between two distributions, and that gives some intuition as to why in it's used in variational methods. Variational methods use functionals as measures in its objective function (i.e. entropy of a distribution takes in a distribution and return a scalar quantity). It's interpreted as the "loss of information" when using one distribution to approximate another, and is desirable in machine learning due to the fact that in models where dimensionality reduction is used, we would like to preserve as much information of the original input as possible. This is more obvious when looking at VAEs which use the KL divergence between the posterior q and prior p distribution over the latent variable z. Likewise, you can refer to EM, where we decompose

ln p(X) = L(q) + KL(q||p)

Here we maximize the lower bound on L(q) by minimizing the KL divergence, which becomes 0 when p(Z|X) = q(Z). However, in many cases, we wish to restrict the family of distributions and parameterize q(Z) with a set of parameters w, so we can optimize w.r.t. w.

Note that KL(p||q) = - \sum p(Z) ln (q(Z) / p(Z)), and so KL(p||q) is different from KL(q||p). This asymmetry, however, can be exploited in the sense that in cases where we wish to learn the parameters of a distribution q that over-compensates for p, we can minimize KL(p||q). Conversely when we wish to seek just the main components of p with q distribution, we can minimize KL(q||p). This example from the Bishop book illustrates this well.

KL divergence belongs to an alpha family of divergences, where the parameter alpha takes on separate limits for the forward and backwards KL. When alpha = 0, it becomes symmetric, and linearly related to the Hellinger distance. There are other metrics such as the Cauchy Schwartz divergence which are symmetric, but in machine learning settings where the goal is to learn simpler, tractable parameterizations of distributions which approximate a target, they might not be as useful as KL.