Markdown vs markup - are they related?

I'm using markdown to edit this question right now. In some wikis I used wiki markup. Are they the same thing? Are they related? Please explain. If I want to implement one or the other in a web project (like stackoverflow) what do I need to use?

• Markup is a generic term for a language that describes a document's formatting
• Markdown is a specific markup library: http://daringfireball.net/projects/markdown/
  These days the term is more commonly used to refer to markup languages that mimic the style of the library. See: https://en.wikipedia.org/wiki/Markdown