Its really easy to embed a YouTube video on your website and make sure that 'related' videos don't show up after yours has finished playing. (note: if you have a wordpress site the instructions are slightly different, click here to read more)
Step 1 - Find the YouTube video you want to embed and click on the 'Share' button under it.
Step 2 - Click on the 'Embed' button
Step 3 - Make sure the 'Show suggested videos...' option is un-checked
Step 4 - Copy the code and paste it where you want it on your website.