There are several approaches for doing Bayesian A/B testing.
First of all, you should decide whether you want to use an analytic approach (using conjugate distributions as Lenwood mentions) or an MCMC approach. For simple A/B experiments, particularly on conversion rate which is your case, there is really no need to use an MCMC approach: just use a Beta distribution as a prior and your posterior distribution will also be a Beta distribution.
Then, you need to decide which decision rule to apply. Here, there seems to be two main approaches for decision making. The first one is based on a paper by John Kruschke from Indiana University (K. Kruschke, Bayesian Estimation Supersedes the t Test, Journal of Experimental Psychology: General, 142, 573 (2013).). The decision rule used in this paper is based on the concept of Region Of Practical Equivalence (ROPE).
Another possibility is to use the concept of an Expected Loss. It has been proposed by Chris Stucchio (C. Stucchio, Bayesian A/B Testing at VWO).
In principle, you could use a different decision rule.
You can find this and much more on this blog post: Bayesian A/B Testing: a step-by-step guide. It also includes some Python code snippets and uses a Python project that is hosted on Github.