See also: webmention, the plumbing used to notify posts that they’ve been linked to
A mention is not really a “type” of post from the publisher’s perspective (i.e. there is no “mention” UI), as any sort of post which can link can implicitly be a mention, as well as potentially being in response to another post.
Mentions typically don’t fit into comment/reply threads and therefore are often displayed separately, or in a more compact manner.
Unlike a like, a mention is not implicitly emotionally positive — a link is not always an endorsement.
IndieWeb examples displaying mentions to posts
Aaron displays mentions to posts on aaronparecki.com in an “other mentions” section below likes, reposts and comments, as well as giving the mention count in feed views (example).
Barnaby displays mentions to posts on waterpigs.co.uk inline chronologically with other responses, but in a more compact form that explicit comments, and displays mention counts in feed views (example).
Examples of Mastodon doing completion support (instance-local and federated-wide):
Noted above, @AreOh is local to the instance that @jalcine is on. However, everything else below are people that @jalcine follows.
Mention Versus Reply
A webmention get sent when you link to another website. A reply is a specific post that requires markup on the link to indicate that someone is publishing a reply rather than a generic mention.
A key difference is the overall goal of the post. If the purpose of al the content is to reply to content somewhere else on the web then use reply.
If the intent is to use a link in an article where the reply is in support of the purpose of the post, rather than the purpose of the post itself use a mention