When using video content you did not create, for example YouTube videos, you must keep fair use in mind. If material you are using is copyrighted you need permission to use it from the copyright holder. There are four factors to consider in making a fair use decision:
1. The purpose and character of your use. The fact that your use is nonprofit and educational helps the fair use argument.
2. The nature of the original. If the original work is nonfiction/factual in nature, there’s a better argument for fair use than if the original is creative/fictional. It also helps the fair use argument if the original was published.
3. The amount you’re using. Here you’re looking at the proportion of the whole original work you want to use. There is no definite amount that is or isn’t okay, but the shorter the clips you’re using, the better. You should be using no more than is necessary to make your pedagogical point.
4. Effect on the market for the original. If your use is simply replacing a sale of the original, the fair use argument is less strong. Of course the less you use, the less likely it is to interfere with the market for the original.
All of these factors don’t have to favor fair use in order for your use to be fair - it’s more of a risk analysis.
In addition to fair use, the TEACH Act also enables for some displays of “reasonable and limited” portions of videos during online classes. In general you should be able to play short clips during class that have a definite pedagogical purpose, as long as you’re sensitive to the market effects of the use. The TEACH Act only enables you to incorporate clips that you would normally play during the course of a face-to-face class, not what you’d be expecting students to view on their own time outside of class.