As documented in WordPress trac ticket #20102, there is an issue with oEmbeds from some services which have started allowing/using https URLs rather than http URLs. The ticket documents an issue with the oEmbed functionality, oEmbed is the technology which allows you to paste a YouTube URL into your blog and have that converted to an embedded YouTube video when someone looks at the post. The problem is that the list of “safe services to embed from” does not account for URLs prefixed with the secure https protocol (as opposed to the non-secure http which is used for most web traffic which simply does not need the security).
The solution from ticket #20102 should be included in the next release of WordPress (hopefully), but in the meantime you can use this quick plugin I wrote which implements the new providers list, I’m waiting on WordPress.org plugin approval but in the meantime here’s the Github URL: HTTPS oEmbed Providers on Github
Download the files as a “zip”, unzip on your computer, then upload to your WordPress plugins directory; or upload the zip through the plugins page in your WordPress admin area.
The plugin is designed to warn you when the next version of WordPress is released, when hopefully you can deactivate and delete it.
The plugin only enables HTTPS for the providers which are already listed by WordPress, it does not add new providers (i.e. if you want an extra oEmbed provider then you need to look elsewhere, sorry).
(Post updated after publication to, hopefully, make the problem clearer.)