If you mean the search history in the Google Toolbar (if you have it), just click the down arrow next to the Google button, on the left of the toolbar, and select Clear Search History.
If you mean the search history on the normal Google search engine, that's part of Internet Explorer's AutoComplete. You can try selectively highlight an item from the history and press the delete button on the keyboard.
Or to remove everything, in IE, click Tools, Internet Options, Content tab, AutoComplete, Clear forms - note this will clear all forms saved by Internet Explorer.
You can also disable AutoComplete here. If you disable this type of AutoComplete (which gives a drop down list), there is still an alternative to use, (which gives the closest match from your history on the line itself without showing you all the near matches in the drop down list - a little more "private").
You can enable the alternative by going to Advanced tab, tick "Use in-line AutoComplete".
Or, don't use AutoComplete at all.
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