Difference between BD-R, BD-RE, DVD-R, DVD+R

If you are interested in getting a DVD or Blu-ray drive to burn some data off to an external disc that you can store in a safe deposit box or for some other reason, then the sheer number of formats and versions can make the task quite confusing. For example, there are five different formats

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What's the Difference Between DVD+R and DVD-R?

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DVD-R VS DVD+R: Difference between DVD-R and DVD+R