Eric is absolutely right. But you can "trick" the Content Editor Web Part into displaying a background color. Using this Each Content editor web part can have a different background color.
Basically you apply a table cell to the entire content editor web part and change the cell color.
Open the Content Editor Menu (Modify Shared Web Part)
Click Source Editor
Note: Do not save until the top and bottom code has been added
At the VERY TOP before any other code paste in the following code:
<TABLE height="100%" width="100%">
<TR>
<TD bgColor="lightblue">
Now at the VERY BOTTOM after all other code paste in the following code:
</TD>
</TR>
</TABLE>
Click Save.
You can modify the color that displays by adding the color name or Hex# after bgColor=. You can still use the Rich Text Editor as normal after adding the code.
For information about colors you can use and different hex codes (Example-#FFFFFF) I recommend Web Monkey - http://www.webmonkey.com/reference/Color_Charts
Hope that helps.