On my page I have a content div that only shows when the mouse hovers over the sidebar div.
How do I remove only the border where the 2 divs touch like the picture below shows?

I need it to work in chrome, firefox & IE 8+
Here is a jsfiddle that sets up an example of my issue: http://jsfiddle.net/paulyoder/K2wKG/
For simplicity, please provide a jsfiddle of your solution
Declare a background color of white for Side Bar
Declare a border-right: none; for Side Bar
then have side bar over lap Content by 1px this last step can be done may ways but I would need to see your specific markup.
Margin-left: -1px for content may do the trick!
Hope this helps
Here is a fiddle doing exactly as you describe. http://jsfiddle.net/3E4mD/1/
In the future it would be easier to see what you have and advise from there.
Thanks