In the beginning, there was the markdown.pl perl script (probably still available on gruber's blog). It converted a markdown file to an HTML file. It was probably also available as a wordpress plugin, which would convert markdown text in the textarea box to HTML.
That pretty much covers Markdown to HTML conversion today - either a standalone utility (markdown.pl, lowdown, commonmark, etc) or a library for your preferred language (kramdown, cmark, etc).
I guess the only other advancement is markdown-savvy editors that render the HTML in realtime as you type.