Quick answer: Simply add '?rel=0' after the video ID.
For example, you want to embed our latest animation showreel on your website just cos you think its awesome. You've grabbed this code from the YouTube video page:
<iframe width="560" height="315" src="//www.youtube.com/embed/PnaQFTU4_c8" ></iframe>
Look for the unique YouTube code in the URL, and simply add ?rel=0 after it like so:
<iframe width="560" height="315" src="//www.youtube.com/embed/PnaQFTU4_c8?rel=0" ></iframe>
If you're like most people and have no idea about HTML or coding, YouTube have also made it easy for you. And I've summarised it here below: