NOTE: The techniques outlined below require switching between web pages, so it's worthwhile knowing that in Windows a "wheel-click" opens a link in a new tab, rather than replacing the contents of the current tab with the new page. Having a number of tabs open makes switching between pages that are to be the target of links and embeddings so much simpler with much less danger of half completed posts being lost! If you just left-click links then you are likely to lose anything you type on a page by the time you return to it!
1. Visit the YouTube video you wish to embed.
2. Click on the SHARE button

3. Click on the EMBED button and do what the dialogue says!

(That means select the options you want - I'd recommend removing the default "Show suggested videos" and setting the video size at at least 640x510 - and then copy - CTRL-C - the selected text)
4. Switch to the message you are writing and click the TIPS button

5. Click the EMBED button at the bottom of the dialogue. This inserts the special "tags" into the EDIT box:
6. Place the cursor between the two tags and hit CTRL-V

You have now added the code necessary to embed the video.
Before you finish editing your message, it's worth clicking the "Preview Message" button to check that everything is working as expected.