First, make sure you are using a Child Theme to make your CSS edits.
I agree with Ross that you'll want to look into using the browser plugin Firebug, using that you can directly see and test CSS in the browser.
The Twenty Twelve footer background can be changed like this, but you'll most likely want to make some additional changes as the footer does not stretch 100% across the screen due to some padding.
footer[role="contentinfo"] {
background: none repeat scroll 0 0 #eeeeee;
}