File extension for Markdown files?

There is no requirement for a Markdown file extension, as other answers have explained. But in order for editors or parsers to guarantee that the file they are using is Markdown-formatted, they would look for one of the following extensions:

.markdown
.mdown
.mkdn
.md
.mkd
.mdwn
.mdtxt
.mdtext
.text
.Rmd

There are websites such as GitHub that only use a selection of these extensions for converting to HTML so developers will conform to their standard. (see examples below)

Personally, I have seen .markdown and .mdown used the most, and as a Linux user I would avoid using .md as this can also be a machine description file for compiling code with GCC.

Examples of extension usage:

GitHub: markdown, mdown, mkdn, mkd, md (source)

Elements Markdown Editor: markdown, mdown, mdwn, md

Vim markdown: markdown, mdown, mkdn, mdwn, mkd, md

Bitbucket: markdown, mdown, mkdn, mkd, md, text (source)

R Studio: Rmd

Further Reading

There is a Markdown mailing list that has interesting discussions about this topic: 1, 2.

And especially revealing is one of the explanations:

Markdown isn't meant to take over the format of a file, it's a way to subtly add information to the plaintext. Really, the presence of Markdown is metadata, not a file format.
...
No one opening a text file will be confused if they find Markdown syntax, it's pure bonus.

In this sense, it makes sense to use ".text", ".txt", or whatever other plaintext extension is relevant.
...
An editor which knows nothing about Markdown won't care about the metadata and won't be confused by the variety of "non-standard" extensions, but will display and edit the plaintext just fine.

The idea of MarkDown and similar light-weight markups is to be readable as plain text, thus they have text/plain extension (.txt or .text).

However, there are some people who use .markdown or .mdown.

I'm not aware of one, but I think a precedent is set by the use of the .text extension on the official website to reveal the Markdown that produces the pages.

iA Writer uses the extension .md for Markdown content.