Please help me delete YouTube videos that show up when I Google myself. They are old embarrassing videos and I don't want anyone to see them when they Google my name.
Have you contacted the person who posted the video? You could try to convince them into removing the video.
Otherwise, depending on where you are in the world, you might be able to claim the Right to be forgotten.
Under the offending video, click More → Report:
This will give you a number of options for why you want to report the video. You probably want to go with something related to Privacy, but it is up to YouTube if they want to remove the video or not.