How do people use copyrighted music in videos without it being taken down?

DISCLAIMER: I make videos for nonprofits and for my own artistic outlet, they get very little views, and I do not attempt to monetize these videos in any way.

How do I find out what songs I am allowed to use in YouTube videos? I know that the most obvious answer is any copyright-free music, but I have used songs (like Pretty Bones by Yeule, for example) in videos before and they got by fine, just weren't monetizable, of course. I have to make a video for a nonprofit soon under a deadline, so the last thing I need is to finish the video and have it denied as I attempt to upload it. How do you know what songs you can and can't use??