A/B testing is all the rage nowadays, but not everybody gets it yet. Let's investigate!

Here are a number of introductory presentations and resources which will help you get the basic idea of what it is all about.

Animated presentation on ‘What is A/B Testing’.

The Simplest A/B Testing Guide

And here is a great introductory article:

https://learn.optimizely.com/hc/en-us/articles/201921737-Strategy-Lesson-Introducing-A-B-Testing-and-Optimization

Split-testing basics

Here is a nice video on the basics of split-testing:

Case-studies

The website of Visual Web Optimizer (one of the tools, used for A/B testing) include a broad range of case-studies, explaining the merits of A/B testing in business context:

https://vwo.com/ideafox/

There quite a number of case studies and user stories on the website of another very popular A/B testing tool Optimizely:

http://blog.optimizely.com/category/casestudies/

https://www.optimizely.com/customers/customer-stories