You should place the images in your template folder not the wp-admin folder. Usually the best place for images is “wp-content/templates/your-template-name/images(or other folder name where the images are)” and than in the CSS reference a relative path: {background-image: url (images/your-image.jpg)} without the first slash. If you place your images in wp-admin, they will be removed by the system at each system update
If you want to keep them in the wp-admin anyway (what I would discourage), you should give a path starting from the folder in which your wp instalation resides, not from the top folder on your server. In your case, this would be the public_html, so your code should look like this: {background-image: url (/wp-admin/images/your-image.jpg);} – the whole path goes into the brackets and you place url before the brackets.

I hope this will help.

.blockquote {background-image: /home/tallrorg/public_html/wp-admin(images/Open Quotes.jpg); background-repeat: no-repeat; background-position: left 5px; padding-left: 30px; font-style: italic;}
.blockquote span {background-image: /home/tallrorg/public_html/wp-admin(images/Close Quotes.jpg); background-repeat: no-repeat; background-position: right bottom; display: block; padding-right: 30px;}

For some reason though, it wasn’t pulling the images on the site. I made the right HTML references also. Do you know why this is happening? What did I do wrong here?