I share a computer but not my screen. I have my own screen names.
How can I erase my search history w/o erasing his?

You will easily be able to delete your history by clicking on History in your brower's menu without erasing another user's history. The exact menu items you must click to erase your history depends upon the browser you are using. What browser are you using?

AOL broadband

That's a service provider... not the browser.