Hi Sheryl.
Based upon your explanations, I have a couple of thoughts.
The folder for your images can be called anything, so Backgrounds or Backgrounds_html should work equally well.
I suspect that your trouble is that Linux is case-sensitive. So if your folder is named Backgrounds, the references should be Backgrounds/graphic.jpg rather than backgrounds/graphic.jpg.
Now, again about the reference in the code: using Backgrounds/graphic.jpg produces a relative reference, meaning that the browser will look for the Backgrounds folder in the folder containing the html page itself. To reference the Backgrounds folder from anywhere, and allow portability of your html, you can use /Backgrounds/graphic.jpg, which looks for the Backgrounds folder directly under public_html, no matter where the current page lives.
Hope that helps a bit.