When you choose to embed a YouTube video, you have the option of displaying related videos and a search bar within the embed window. You can choose not to display related videos by selecting that option when creating the embed code within YouTube.
If you've already embedded the video but want to disable the related videos, you'll need to change the code. If it all looks like gibberish to you, don't worry. Just find the part of the code that reads rel=1 and change it to rel=0.