In general, you are fine to quote from copyrighted texts with proper attribution. Keep in mind that you should use only the amount of text necessary to support your argument or conduct your own analysis; this is good scholarship and best practice in adhering to copyright law. If you are conducting research in an archive or have access to unpublished texts, a good resource is the Society of American Archivists "Copyright and Unpublished Material" guide.
As you incorporate images or video clips into your work, ask the following questions to decide whether or not you need to get permission:
Is the work in copyright? If you're using a video or recent image created within the past 40 years, it is very likely protected by copyright. If you're using older material, it may or may not be protected. This chart is a great starting point when deciding whether or not the image is copyrighted.
If it is protected, can you make the case for fair use? Fair use is a part of U.S. Copyright Law that supports limited uses of copyrighted materials for education and scholarship. In the context of theses and dissertations, you may be able to rely on fair use rather than obtaining specific permission from the copyright holder. As one way to assess this, ask yourself whether or not the image or video is necessary to your argument? For instance, are you analyzing the work in your writing or does it directly support a particular point you are trying to make? If yes, your use is more likely to be fair. If the image or video is mainly there to make your work more visually appealing, but without really adding anything crucial to your argument, that use is less likely to be fair.
Can you use an image that is in the public domain or royalty free? If you don't think your use of a specific image falls within fair use, consider using a free-to-use alternative. Most images created before the 20th century and many images created after that are in the public domain, meaning their copyright protections have expired or they were never protected in the first place. Many websites also post images labeled as "royalty free" or "openly licensed." Places to start include CC Search, Unsplash, and Wikipedia's list of public domain image resources.
Do you need permission? If you want to use a specific image or video, have determined your use isn't fair, and can't find a suitable alternative, you may be able to seek permission from the copyright holder or pay a licensing fee. If you know the name of the photographer, you may be able to find their contact information online. On YouTube, you can typically send a message by going to a user's "about" page (though keep in mind that many people upload videos without owning the copyright). Note that most archives and special collections do not own copyright to many of the materials that they own; staff at such institutions may be able to give you information about the creator but often cannot give you permission to use the item.
Similarly, you may typically republish a chart or graph that conveys factual information in a straightforward, uncreative way (e.g. a simple bar graph, pie chart, etc.). If the chart or graph involves more creative design or infographic elements, annotation, etc., you will need to consider fair use, get permission from the publisher or author, or create your own illustration based on the underlying facts.