Thanks for your question. I am assuming you are referring to the
search box at Google.com. If you are referring to the search history
on the Google Tool Bar, I will address that at the end of my answer.
While it may not seem like a feature, the cause of your concerns is
known as AutoComplete. It is designed to make frequently searched
terms more easily accessible for future searches. However, in your
case, you don't want those terms available. There are a couple of
ways to rectify your situation.
In Internet Explorer, if you go to the Tools menu, then select
Internet Options..., then click on the Content tab, you will see a
section called Personal Information. The first button in this section
is the AutoComplete feature. When you click this button, another set
of options will appear. To ensure that Explorer no longer remembers
any future searches, deselect (uncheck) "Forms". Now, bear in mind,
this will affect ALL forms you may fill, not just the Google search
box. Next, click "Clear Forms". This will remove the entries stored
from previous searches.
If there are only certain search terms you want to remove, you can go
to Google and double click in the search form. A drop down list of
all of the stored search terms will appear. Clicking your down arrow
will highlight the first term. You can then hit Delete to delete that
term or use your down arrow to move to the next term, delete if
wanted, and so on.
If you are referring the to the search history stored in the Google
Toolbar, to clear this, click the Google button, then Clear Search
I hope this helps. If you need any additional clarification, please let me know.