Beta diversity quantifies the degree of similarity between groups. However, measure that similarity most commonly in terms of species richness, phylogenetic diversity, species abundance, or functional trait diversity. Most analyses will supply a dissimilarity value between 0 and 1, where 1 indicates they are completely dissimilar from one another (no overlapping diversity), while a 0 indicates they are completely similar (100% overlapping diversity). 

We can use Beta diversity to tell us how similar two species assemblages are to one another. Which may be useful to detect changes in experimental vs control groups, across habitats, or before and after some major event. The average beta diversity may indicate how diverse the system is as a whole, as higher values of beta diversity may indicate more differentiation between groups. 

However, there are many metrics which measure Beta diversity. 

The earliest beta diversity from Whittaker 1960 was simply the local diversity divided by the global diversity, however this has largely fallen out of use. https://esajournals.onlinelibrary.wiley.com/doi/10.2307/1943563

The choice of which beta diversity metric to use often falls on the assumptions of the system and research question. This paper reviews the various metrics, how they behave under scenarios, and provides a great deal of insight into how these various metric operate. https://besjournals.onlinelibrary.wiley.com/doi/full/10.1046/j.1365-2656.2003.00710.x

Papers: 
Great paper that covers the math - https://nsojournals.onlinelibrary.wiley.com/doi/epdf/10.1111/j.1600-0587.2009.05880.x

A good review on empirical biases - https://esajournals.onlinelibrary.wiley.com/doi/10.1002/ecs2.2100