There isn’t a feature in Evernote to add a YouTube video to a note. Pasting the link in doesn’t embed it as it would in another tool like Evernote, either. But there is a workaround. You can use the Evernote Chrome Extension to insert a YouTube video in Evernote. This saves it as a Web Clip in a new note. You can then copy and paste the Web Clip into an existing note. This works on mobile too through the Evernote application for Android and iOS, using your device’s built-in Share feature.
Unfortunately, this isn’t the cleanest solution. Web Clips for YouTube have a sizeable, obtrusive border in Evernote and bring in a lot of excess information that can’t be trimmed out, like the video description, and the content can’t be watched within Evernote, so it isn’t the same as embedding a video in the middle of a note, but outside of including the link to the video or downloading it from YouTube and uploading it as a video file, it’s the next best thing — until official support is added.
Don’t be fooled by the Simplify & Make Editable tool that’s shown next to the Web Clip either. While this will let you remove all of the additional information and the border, it’ll remove the embed turning what was a hyperlink to the external video. What you’ll be left with is an image of the thumbnail. Clicking it won’t redirect you to the video, so it’s a roundabout way to add an image to the note. You’ll need to suffer with the border and excess data if you want it to be a clickable asset to the video.