There are some images you never need permission to use (but your professors might still want you to cite them):
Public Domain images are old enough that the copyright has expired. Some images, like many published by the U.S. Government are put immediately into the public domain, so there's no waiting for copyright to expire. Artists can also dedicate their work into the public domain, for example by assigning it a "Creative Commons Zero" or "CC0" license giving up their own rights to the image.
If you find an image that isn't in the Public Domain, it's likely copyrighted, meaning that all rights are reserved, and only the original creator can publish it.
You might still be able to use a copyrighted image, if you can prove that you aren't preventing the creator from making money. To determine whether your use of an image is "fair use", you can use a Fair Use Checklist, and you should keep a copy of your checklist/thought process in case the creator sends you a "cease and desist" letter or tries to sue you. Some examples of arguments in favor of "fair use":
Some red flags that your use isn't fair:
! These are just examples, librarians can't provide legal advice, so make sure you go through a Checklist and make the determination yourself !
If you want to use an image that isn't in the public domain, meaning that it's copyrighted by someone else, and you don't think your use of the image would fall under "fair use", then you can reach out to the rights-holder (usually the original creator, or a publishing company) to request written permission.
You might find an image that tells you exactly how you're allowed to use it. For example in the description of a YouTube video there might be a statement that says "free to use with credit to the owner", so you could screenshot that statement for your records. There's a more formal way for people to tell you exactly how you're allowed to use their image, called a Creative Commons license:
These symbols mean you can use it as long as you give permission:
These symbols mean you can use it, but only for non-commercial uses: