How can I change the table color of the checkout?

There is nothing to add.
You would have to remove/edit the
#wrapper{
width: 970px;
}

CSS code that you have there.

I removed it and what happened was that the entire site moved was aligned left in the browser. Am I right that I need to change the code in the pure_white template main.css file? Or is it elsewhere to be changed?

The problem is that your template uses the same CSS file for both the main display and the popup display, which should not be the case, specially because of such problems where this CSS is needed for the integrity of the main display and breaks the display when displaying popups. The solution might be to remove the CSS file the component.php file of your template, but you should ask your template provider as it might be more complicated to do...