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You can embed Mastodon posts in your website or blog by doing the following:
1. Log in through your server's website
2. Go to the post or reply you want to embed and click on it to show it in an enlarged view
3. Click ⋯ on the post and select "Get Embed Code"
4. Copy-paste the HTML code into your website or blog
Lots of common questions answered about this at:
https://fedi.tips/how-to-embed-mastodon-posts-on-a-website
If you have problems embedding, see the article for suggestions.
p.s. UPDATED: It seems that there are bugs on this feature when trying to embed some posts that are on other servers.
I've reported this as a bug including examples of it happening, the report is at https://github.com/mastodon/mastodon/issues/34181
I'd mistakenly thought it was just mastodon.social with the problem, thanks to @Stefan_S_from_H for correcting me 
@FediTips It doesn't seem to have anything to do with nightly. It works on some posts and not on others.
I cannot get it to work on any posts on mastodon.social, they all have the same error.
However, it does work on all posts on my own instance which is running release.