Your computer is storing each search you do, as a way to speed up the search in the future. Each type of software, and each operating system is slightly different about how to change this, but generally this is what you need to do:
On many browsers, (Your browser is probably Internet Explorer) you go to a section called "Tools"; After tools, go to "Internet Options"; See if you have an option for "delete history". If so, that's the button you need to hit. After erasing your history, close the browser completely, and then re-open it. Now typing won't show the places you have been. You would have to do this each time you after you've visited a site you don't want to show up in the list, however.
On some browsers, you go to "history" in your browser. There is usually a button of some sort that says "History." Without knowing exactly what operating system, what browser, and what tools you have on your machine I can't be sure. If "tools" doesn't work, try "history." Look around for something that allows your history (or footprints) to be deleted.
Finally, you might look (again in your browser) for cookies, and "delete cookies." Some browsers store such information in a "cookie."