ViewPure is one of my longest standing recommendations for viewing YouTube videos without distractions. At its basic level to use View Pure just copy the link of a video into the "purifier" on the View Pure website and then click purify. Your "purified" video will be displayed on a blank white background. You can password-protect links to videos that you share through ViewPure (click here for directions). In the last year ViewPure expanded to offer curated collections of educational videos.
Watchkin is a service that provides a few ways to watch YouTube videos without seeing the related video suggestions and comments. You can enter the direct URL of a video into Watchkin to have the sidebar content removed. You can search for videos through Watchkin and have family-friendly results displayed (if a video appears that is not family-friendly Watchkin has a mechanism for flagging it as inappropriate). Watchkin also offers a browser bookmarklet tool that you can click while on YouTube.com to have the related content disappear from the page.
Quietube is a convenient tool that you can add to your browser's bookmarks bar. With Quietube installed you can simply click it whenever you're viewing a video on YouTube and all of the related clutter will be hidden from view. Installing Quietube requires nothing more than dragging the Quietube button to your toolbar.
SafeShare.tv makes it possible to view YouTube videos without displaying the related videos and associated comments. To use SafeShare.tv simply copy the URL of a YouTube video and paste it into SafeShare.tv. SafeShare also offers browser a bookmarklet tool that will eliminate the need to copy and paste links from YouTube into SafeShare.
Methods for those who use Google Classroom, Google Sites, or any blogging service:
If you use Google Classroom, you can simply post the link to a video in your Classroom and when students click on it, it should display within Classroom without showing any related content from YouTube.
You can embed YouTube videos into Google Sites, Blogger, and many other website and blog creation platforms. The embedded videos won't display any comments that might have been written on YouTube.